Theorem spwnex 8618

Description: Non-closure when the supremum doesn't exist.

Hypotheses

Ref	Expression
spwval.1	⊢ X = dom R
spwval.2	⊢ (φ ↔ (∀y ∈ A yRx ⋀ ∀y ∈ X (∀z ∈ A zRy → xRy)))

Assertion

Ref	Expression
spwnex	⊢ ((R ∈ Poset ⋀ A ∈ W ⋀ ¬ ∃x ∈ X φ) → ¬ (R supw A) ∈ X)

Distinct variable groups:   x,y,z,A   x,R,y,z   x,X,y

Proof of Theorem spwnex

Step	Hyp	Ref	Expression
1		eqid 1475	. . . . 5 ⊢ ∪∪R = ∪∪R
2		pm4.2 170	. . . . 5 ⊢ ((∀y ∈ A yRx ⋀ ∀y ∈ ∪∪R(∀z ∈ A zRy → xRy)) ↔ (∀y ∈ A yRx ⋀ ∀y ∈ ∪∪R(∀z ∈ A zRy → xRy)))
3	1, 2	spwnex3 8612	. . . 4 ⊢ ((R ∈ Poset ⋀ A ∈ W ⋀ ¬ ∃x ∈ ∪∪R(∀y ∈ A yRx ⋀ ∀y ∈ ∪∪R(∀z ∈ A zRy → xRy))) → ¬ (R supw A) ∈ ∪∪R)
4	3	3expia 835	. . 3 ⊢ ((R ∈ Poset ⋀ A ∈ W) → (¬ ∃x ∈ ∪∪R(∀y ∈ A yRx ⋀ ∀y ∈ ∪∪R(∀z ∈ A zRy → xRy)) → ¬ (R supw A) ∈ ∪∪R))
5		psdmrn 8605	. . . . . . . 8 ⊢ (R ∈ Poset → (dom R = ∪∪R ⋀ ran R = ∪∪R))
6	5	pm3.26d 321	. . . . . . 7 ⊢ (R ∈ Poset → dom R = ∪∪R)
7		spwval.1	. . . . . . 7 ⊢ X = dom R
8	6, 7	syl5eq 1518	. . . . . 6 ⊢ (R ∈ Poset → X = ∪∪R)
9		raleq1 1785	. . . . . . . . 9 ⊢ (X = ∪∪R → (∀y ∈ X (∀z ∈ A zRy → xRy) ↔ ∀y ∈ ∪∪R(∀z ∈ A zRy → xRy)))
10	9	anbi2d 616	. . . . . . . 8 ⊢ (X = ∪∪R → ((∀y ∈ A yRx ⋀ ∀y ∈ X (∀z ∈ A zRy → xRy)) ↔ (∀y ∈ A yRx ⋀ ∀y ∈ ∪∪R(∀z ∈ A zRy → xRy))))
11		spwval.2	. . . . . . . 8 ⊢ (φ ↔ (∀y ∈ A yRx ⋀ ∀y ∈ X (∀z ∈ A zRy → xRy)))
12	10, 11	syl5bb 532	. . . . . . 7 ⊢ (X = ∪∪R → (φ ↔ (∀y ∈ A yRx ⋀ ∀y ∈ ∪∪R(∀z ∈ A zRy → xRy))))
13	12	rexeqd 1791	. . . . . 6 ⊢ (X = ∪∪R → (∃x ∈ X φ ↔ ∃x ∈ ∪∪R(∀y ∈ A yRx ⋀ ∀y ∈ ∪∪R(∀z ∈ A zRy → xRy))))
14	8, 13	syl 10	. . . . 5 ⊢ (R ∈ Poset → (∃x ∈ X φ ↔ ∃x ∈ ∪∪R(∀y ∈ A yRx ⋀ ∀y ∈ ∪∪R(∀z ∈ A zRy → xRy))))
15	14	negbid 611	. . . 4 ⊢ (R ∈ Poset → (¬ ∃x ∈ X φ ↔ ¬ ∃x ∈ ∪∪R(∀y ∈ A yRx ⋀ ∀y ∈ ∪∪R(∀z ∈ A zRy → xRy))))
16	15	adantr 389	. . 3 ⊢ ((R ∈ Poset ⋀ A ∈ W) → (¬ ∃x ∈ X φ ↔ ¬ ∃x ∈ ∪∪R(∀y ∈ A yRx ⋀ ∀y ∈ ∪∪R(∀z ∈ A zRy → xRy))))
17	8	eleq2d 1540	. . . . 5 ⊢ (R ∈ Poset → ((R supw A) ∈ X ↔ (R supw A) ∈ ∪∪R))
18	17	negbid 611	. . . 4 ⊢ (R ∈ Poset → (¬ (R supw A) ∈ X ↔ ¬ (R supw A) ∈ ∪∪R))
19	18	adantr 389	. . 3 ⊢ ((R ∈ Poset ⋀ A ∈ W) → (¬ (R supw A) ∈ X ↔ ¬ (R supw A) ∈ ∪∪R))
20	4, 16, 19	3imtr4d 543	. 2 ⊢ ((R ∈ Poset ⋀ A ∈ W) → (¬ ∃x ∈ X φ → ¬ (R supw A) ∈ X))
21	20	3impia 830	1 ⊢ ((R ∈ Poset ⋀ A ∈ W ⋀ ¬ ∃x ∈ X φ) → ¬ (R supw A) ∈ X)

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 146   ⋀ wa 223   ⋀ w3a 775   = wceq 956   ∈ wcel 958  ∀wral 1644  ∃wrex 1645  ∪cuni 2501   class class class wbr 2617  dom cdm 3168  ran crn 3169  (class class class)co 3961  Posetcps 8590   sup_w cspw 8591

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2691  ax-sep 2701  ax-pow 2740  ax-pr 2777  ax-un 2864

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-rex 1649  df-rab 1651  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-if 2360  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-uni 2502  df-br 2618  df-opab 2665  df-id 2833  df-xp 3182  df-rel 3183  df-cnv 3184  df-co 3185  df-dm 3186  df-rn 3187  df-res 3188  df-ima 3189  df-fun 3190  df-fn 3191  df-f 3192  df-f1 3193  df-fo 3194  df-f1o 3195  df-fv 3196  df-opr 3963  df-oprab 3964  df-er 4259  df-en 4365  df-dom 4366  df-sdom 4367  df-ps 8596  df-spw 8597

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