Theorem supmax 4589

Description: The greatest element of a set is its supremum. Note that the converse is not true; the supremum might not be an element of the set considered. (Contributed by Jeff Hoffman, 17-Jun-2008.)

Hypothesis

Ref	Expression
supmax.1	⊢ R Or A

Assertion

Ref	Expression
supmax	⊢ ((C ∈ A ⋀ C ∈ B ⋀ ∀y ∈ B ¬ CRy) → sup(B, A, R) = C)

Distinct variable groups:   y,A   y,B   y,C   y,R

Proof of Theorem supmax

Step	Hyp	Ref	Expression
1		supmax.1	. . . . 5 ⊢ R Or A
2	1	supub 4580	. . . 4 ⊢ (∃x ∈ A (∀z ∈ B ¬ xRz ⋀ ∀z ∈ A (zRx → ∃y ∈ B zRy)) → (C ∈ B → ¬ sup(B, A, R)RC))
3		supmaxlem 4588	. . . 4 ⊢ ((C ∈ A ⋀ C ∈ B ⋀ ∀y ∈ B ¬ CRy) → ∃x ∈ A (∀z ∈ B ¬ xRz ⋀ ∀z ∈ A (zRx → ∃y ∈ B zRy)))
4		3simp2 789	. . . 4 ⊢ ((C ∈ A ⋀ C ∈ B ⋀ ∀y ∈ B ¬ CRy) → C ∈ B)
5	2, 3, 4	sylc 68	. . 3 ⊢ ((C ∈ A ⋀ C ∈ B ⋀ ∀y ∈ B ¬ CRy) → ¬ sup(B, A, R)RC)
6	1	supnub 4582	. . . 4 ⊢ (∃x ∈ A (∀z ∈ B ¬ xRz ⋀ ∀z ∈ A (zRx → ∃y ∈ B zRy)) → ((C ∈ A ⋀ ∀y ∈ B ¬ CRy) → ¬ CRsup(B, A, R)))
7		3simpb 786	. . . 4 ⊢ ((C ∈ A ⋀ C ∈ B ⋀ ∀y ∈ B ¬ CRy) → (C ∈ A ⋀ ∀y ∈ B ¬ CRy))
8	6, 3, 7	sylc 68	. . 3 ⊢ ((C ∈ A ⋀ C ∈ B ⋀ ∀y ∈ B ¬ CRy) → ¬ CRsup(B, A, R))
9	5, 8	jca 288	. 2 ⊢ ((C ∈ A ⋀ C ∈ B ⋀ ∀y ∈ B ¬ CRy) → (¬ sup(B, A, R)RC ⋀ ¬ CRsup(B, A, R)))
10		sotrieq2 2862	. . . 4 ⊢ ((R Or A ⋀ (sup(B, A, R) ∈ A ⋀ C ∈ A)) → (sup(B, A, R) = C ↔ (¬ sup(B, A, R)RC ⋀ ¬ CRsup(B, A, R))))
11	1, 10	mpan 695	. . 3 ⊢ ((sup(B, A, R) ∈ A ⋀ C ∈ A) → (sup(B, A, R) = C ↔ (¬ sup(B, A, R)RC ⋀ ¬ CRsup(B, A, R))))
12	1	supcl 4579	. . . 4 ⊢ (∃x ∈ A (∀z ∈ B ¬ xRz ⋀ ∀z ∈ A (zRx → ∃y ∈ B zRy)) → sup(B, A, R) ∈ A)
13	3, 12	syl 10	. . 3 ⊢ ((C ∈ A ⋀ C ∈ B ⋀ ∀y ∈ B ¬ CRy) → sup(B, A, R) ∈ A)
14		3simp1 788	. . 3 ⊢ ((C ∈ A ⋀ C ∈ B ⋀ ∀y ∈ B ¬ CRy) → C ∈ A)
15	11, 13, 14	sylanc 471	. 2 ⊢ ((C ∈ A ⋀ C ∈ B ⋀ ∀y ∈ B ¬ CRy) → (sup(B, A, R) = C ↔ (¬ sup(B, A, R)RC ⋀ ¬ CRsup(B, A, R))))
16	9, 15	mpbird 196	1 ⊢ ((C ∈ A ⋀ C ∈ B ⋀ ∀y ∈ B ¬ CRy) → sup(B, A, R) = C)

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 146   ⋀ wa 223   ⋀ w3a 775   = wceq 956   ∈ wcel 958  ∀wral 1645  ∃wrex 1646   class class class wbr 2619   Or wor 2839  supcsup 4573

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-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-sep 2703  ax-pow 2742  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  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 1587  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-po 2840  df-so 2850  df-sup 4574

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