Theorem supub 2160

Description: A supremum is an upper bound.

Hypothesis

Ref	Expression
supmo.1	⊢ R Or A

Assertion

Ref	Expression
supub	⊢ (∃x ∈ A (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)) → (C ∈ B → ¬ sup(B, A, R)RC))

Distinct variable group(s):   x,y,z,A   x,R,y,z   x,B,y,z

Proof of Theorem supub

Step	Hyp	Ref	Expression
1		breq2 2066	. . . . 5 ⊢ (w = C → (sup(B, A, R)Rw ↔ sup(B, A, R)RC))
2	1	negbid 463	. . . 4 ⊢ (w = C → (¬ sup(B, A, R)Rw ↔ ¬ sup(B, A, R)RC))
3	2	imbi2d 464	. . 3 ⊢ (w = C → ((∃x ∈ A (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)) → ¬ sup(B, A, R)Rw) ↔ (∃x ∈ A (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)) → ¬ sup(B, A, R)RC)))
4		df-sup 2154	. . . . . . . 8 ⊢ sup(B, A, R) = ∪{x ∈ A∣(∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz))}
5	4	cleqcomi 1105	. . . . . . 7 ⊢ ∪{x ∈ A∣(∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz))} = sup(B, A, R)
6		breq1 2065	. . . . . . . . . . . 12 ⊢ (x = sup(B, A, R) → (xRy ↔ sup(B, A, R)Ry))
7	6	negbid 463	. . . . . . . . . . 11 ⊢ (x = sup(B, A, R) → (¬ xRy ↔ ¬ sup(B, A, R)Ry))
8	7	biraldv 1219	. . . . . . . . . 10 ⊢ (x = sup(B, A, R) → (∀y ∈ B ¬ xRy ↔ ∀y ∈ B ¬ sup(B, A, R)Ry))
9		breq2 2066	. . . . . . . . . . . 12 ⊢ (x = sup(B, A, R) → (yRx ↔ yRsup(B, A, R)))
10	9	imbi1d 465	. . . . . . . . . . 11 ⊢ (x = sup(B, A, R) → ((yRx → ∃z ∈ B yRz) ↔ (yRsup(B, A, R) → ∃z ∈ B yRz)))
11	10	biraldv 1219	. . . . . . . . . 10 ⊢ (x = sup(B, A, R) → (∀y ∈ A (yRx → ∃z ∈ B yRz) ↔ ∀y ∈ A (yRsup(B, A, R) → ∃z ∈ B yRz)))
12	8, 11	anbi12d 476	. . . . . . . . 9 ⊢ (x = sup(B, A, R) → ((∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)) ↔ (∀y ∈ B ¬ sup(B, A, R)Ry ∧ ∀y ∈ A (yRsup(B, A, R) → ∃z ∈ B yRz))))
13	12	reuuni2 1956	. . . . . . . 8 ⊢ ((sup(B, A, R) ∈ A ∧ ∃!x ∈ A (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz))) → ((∀y ∈ B ¬ sup(B, A, R)Ry ∧ ∀y ∈ A (yRsup(B, A, R) → ∃z ∈ B yRz)) ↔ ∪{x ∈ A∣(∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz))} = sup(B, A, R)))
14		supmo.1	. . . . . . . . 9 ⊢ R Or A
15	14	supcl 2159	. . . . . . . 8 ⊢ (∃x ∈ A (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)) → sup(B, A, R) ∈ A)
16	14	supeu 2158	. . . . . . . 8 ⊢ (∃x ∈ A (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)) → ∃!x ∈ A (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)))
17	13, 15, 16	sylanc 361	. . . . . . 7 ⊢ (∃x ∈ A (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)) → ((∀y ∈ B ¬ sup(B, A, R)Ry ∧ ∀y ∈ A (yRsup(B, A, R) → ∃z ∈ B yRz)) ↔ ∪{x ∈ A∣(∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz))} = sup(B, A, R)))
18	5, 17	mpbiri 169	. . . . . 6 ⊢ (∃x ∈ A (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)) → (∀y ∈ B ¬ sup(B, A, R)Ry ∧ ∀y ∈ A (yRsup(B, A, R) → ∃z ∈ B yRz)))
19	18	pm3.26d 258	. . . . 5 ⊢ (∃x ∈ A (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)) → ∀y ∈ B ¬ sup(B, A, R)Ry)
20		breq2 2066	. . . . . . 7 ⊢ (y = w → (sup(B, A, R)Ry ↔ sup(B, A, R)Rw))
21	20	negbid 463	. . . . . 6 ⊢ (y = w → (¬ sup(B, A, R)Ry ↔ ¬ sup(B, A, R)Rw))
22	21	rcla4v 1402	. . . . 5 ⊢ (∀y ∈ B ¬ sup(B, A, R)Ry → (w ∈ B → ¬ sup(B, A, R)Rw))
23	19, 22	syl 12	. . . 4 ⊢ (∃x ∈ A (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)) → (w ∈ B → ¬ sup(B, A, R)Rw))
24	23	com12 13	. . 3 ⊢ (w ∈ B → (∃x ∈ A (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)) → ¬ sup(B, A, R)Rw))
25	3, 24	vtoclga 1387	. 2 ⊢ (C ∈ B → (∃x ∈ A (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)) → ¬ sup(B, A, R)RC))
26	25	com12 13	1 ⊢ (∃x ∈ A (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)) → (C ∈ B → ¬ sup(B, A, R)RC))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196   = weq 797   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  ∃!wreu 1203  {crab 1204  ∪cuni 1919   class class class wbr 2054   Or wor 2059  supcsup 2060

This theorem is referenced by:  supubi 2165  suprub 4513

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-po 2128  df-so 2138  df-sup 2154

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