Theorem supmaxlem 4588

Description: A set that contains a greatest element satisfies the antecedent in supremum theorems. This allows sup(A, B, R) to be used in some situations without the completeness axiom. (Contributed by Jeff Hoffman, 17-Jun-2008.)

Assertion

Ref	Expression
supmaxlem	⊢ ((C ∈ A ⋀ C ∈ B ⋀ ∀z ∈ B ¬ CRz) → ∃x ∈ A (∀y ∈ B ¬ xRy ⋀ ∀y ∈ A (yRx → ∃z ∈ B yRz)))

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

Proof of Theorem supmaxlem

Step	Hyp	Ref	Expression
1		breq1 2622	. . . . . . 7 ⊢ (x = C → (xRy ↔ CRy))
2	1	negbid 611	. . . . . 6 ⊢ (x = C → (¬ xRy ↔ ¬ CRy))
3	2	ralbidv 1663	. . . . 5 ⊢ (x = C → (∀y ∈ B ¬ xRy ↔ ∀y ∈ B ¬ CRy))
4		breq2 2623	. . . . . . 7 ⊢ (x = C → (yRx ↔ yRC))
5	4	imbi1d 613	. . . . . 6 ⊢ (x = C → ((yRx → ∃z ∈ B yRz) ↔ (yRC → ∃z ∈ B yRz)))
6	5	ralbidv 1663	. . . . 5 ⊢ (x = C → (∀y ∈ A (yRx → ∃z ∈ B yRz) ↔ ∀y ∈ A (yRC → ∃z ∈ B yRz)))
7	3, 6	anbi12d 628	. . . 4 ⊢ (x = C → ((∀y ∈ B ¬ xRy ⋀ ∀y ∈ A (yRx → ∃z ∈ B yRz)) ↔ (∀y ∈ B ¬ CRy ⋀ ∀y ∈ A (yRC → ∃z ∈ B yRz))))
8	7	rcla4ev 1877	. . 3 ⊢ ((C ∈ A ⋀ (∀y ∈ B ¬ CRy ⋀ ∀y ∈ A (yRC → ∃z ∈ B yRz))) → ∃x ∈ A (∀y ∈ B ¬ xRy ⋀ ∀y ∈ A (yRx → ∃z ∈ B yRz)))
9		breq2 2623	. . . . . . . 8 ⊢ (z = y → (CRz ↔ CRy))
10	9	negbid 611	. . . . . . 7 ⊢ (z = y → (¬ CRz ↔ ¬ CRy))
11	10	cbvralv 1800	. . . . . 6 ⊢ (∀z ∈ B ¬ CRz ↔ ∀y ∈ B ¬ CRy)
12	11	biimp 151	. . . . 5 ⊢ (∀z ∈ B ¬ CRz → ∀y ∈ B ¬ CRy)
13		breq2 2623	. . . . . . . . 9 ⊢ (z = C → (yRz ↔ yRC))
14	13	rcla4ev 1877	. . . . . . . 8 ⊢ ((C ∈ B ⋀ yRC) → ∃z ∈ B yRz)
15	14	ex 373	. . . . . . 7 ⊢ (C ∈ B → (yRC → ∃z ∈ B yRz))
16	15	a1d 12	. . . . . 6 ⊢ (C ∈ B → (y ∈ A → (yRC → ∃z ∈ B yRz)))
17	16	r19.21aiv 1713	. . . . 5 ⊢ (C ∈ B → ∀y ∈ A (yRC → ∃z ∈ B yRz))
18	12, 17	anim12i 333	. . . 4 ⊢ ((∀z ∈ B ¬ CRz ⋀ C ∈ B) → (∀y ∈ B ¬ CRy ⋀ ∀y ∈ A (yRC → ∃z ∈ B yRz)))
19	18	ancoms 436	. . 3 ⊢ ((C ∈ B ⋀ ∀z ∈ B ¬ CRz) → (∀y ∈ B ¬ CRy ⋀ ∀y ∈ A (yRC → ∃z ∈ B yRz)))
20	8, 19	sylan2 451	. 2 ⊢ ((C ∈ A ⋀ (C ∈ B ⋀ ∀z ∈ B ¬ CRz)) → ∃x ∈ A (∀y ∈ B ¬ xRy ⋀ ∀y ∈ A (yRx → ∃z ∈ B yRz)))
21	20	3impb 829	1 ⊢ ((C ∈ A ⋀ C ∈ B ⋀ ∀z ∈ B ¬ CRz) → ∃x ∈ A (∀y ∈ B ¬ xRy ⋀ ∀y ∈ A (yRx → ∃z ∈ B yRz)))

Syntax hints:  ¬ wn 2   → wi 3   ⋀ wa 223   ⋀ w3a 775   = wceq 956   ∈ wcel 958  ∀wral 1645  ∃wrex 1646   class class class wbr 2619

This theorem is referenced by:  supmax 4589

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-12 968  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

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ral 1649  df-rex 1650  df-v 1812  df-un 2050  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620

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