Theorem suplub 2161

Description: A supremum is the least upper bound.

Hypothesis

Ref	Expression
supmo.1	⊢ R Or A

Assertion

Ref	Expression
suplub	⊢ (∃x ∈ A (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)) → ((C ∈ A ∧ CRsup(B, A, R)) → ∃z ∈ B CRz))

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

Proof of Theorem suplub

Step	Hyp	Ref	Expression
1		breq1 2065	. . . . . 6 ⊢ (w = C → (wRsup(B, A, R) ↔ CRsup(B, A, R)))
2		breq1 2065	. . . . . . 7 ⊢ (w = C → (wRz ↔ CRz))
3	2	birexdv 1220	. . . . . 6 ⊢ (w = C → (∃z ∈ B wRz ↔ ∃z ∈ B CRz))
4	1, 3	imbi12d 474	. . . . 5 ⊢ (w = C → ((wRsup(B, A, R) → ∃z ∈ B wRz) ↔ (CRsup(B, A, R) → ∃z ∈ B CRz)))
5	4	imbi2d 464	. . . 4 ⊢ (w = C → ((∃x ∈ A (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)) → (wRsup(B, A, R) → ∃z ∈ B wRz)) ↔ (∃x ∈ A (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)) → (CRsup(B, A, R) → ∃z ∈ B CRz))))
6		df-sup 2154	. . . . . . . . 9 ⊢ sup(B, A, R) = ∪{x ∈ A∣(∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz))}
7	6	cleqcomi 1105	. . . . . . . 8 ⊢ ∪{x ∈ A∣(∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz))} = sup(B, A, R)
8		breq1 2065	. . . . . . . . . . . . 13 ⊢ (x = sup(B, A, R) → (xRy ↔ sup(B, A, R)Ry))
9	8	negbid 463	. . . . . . . . . . . 12 ⊢ (x = sup(B, A, R) → (¬ xRy ↔ ¬ sup(B, A, R)Ry))
10	9	biraldv 1219	. . . . . . . . . . 11 ⊢ (x = sup(B, A, R) → (∀y ∈ B ¬ xRy ↔ ∀y ∈ B ¬ sup(B, A, R)Ry))
11		breq2 2066	. . . . . . . . . . . . 13 ⊢ (x = sup(B, A, R) → (yRx ↔ yRsup(B, A, R)))
12	11	imbi1d 465	. . . . . . . . . . . 12 ⊢ (x = sup(B, A, R) → ((yRx → ∃z ∈ B yRz) ↔ (yRsup(B, A, R) → ∃z ∈ B yRz)))
13	12	biraldv 1219	. . . . . . . . . . 11 ⊢ (x = sup(B, A, R) → (∀y ∈ A (yRx → ∃z ∈ B yRz) ↔ ∀y ∈ A (yRsup(B, A, R) → ∃z ∈ B yRz)))
14	10, 13	anbi12d 476	. . . . . . . . . 10 ⊢ (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))))
15	14	reuuni2 1956	. . . . . . . . 9 ⊢ ((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)))
16		supmo.1	. . . . . . . . . 10 ⊢ R Or A
17	16	supcl 2159	. . . . . . . . 9 ⊢ (∃x ∈ A (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)) → sup(B, A, R) ∈ A)
18	16	supeu 2158	. . . . . . . . 9 ⊢ (∃x ∈ A (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)) → ∃!x ∈ A (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)))
19	15, 17, 18	sylanc 361	. . . . . . . 8 ⊢ (∃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)))
20	7, 19	mpbiri 169	. . . . . . 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)))
21	20	pm3.27d 262	. . . . . 6 ⊢ (∃x ∈ A (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)) → ∀y ∈ A (yRsup(B, A, R) → ∃z ∈ B yRz))
22		breq1 2065	. . . . . . . 8 ⊢ (y = w → (yRsup(B, A, R) ↔ wRsup(B, A, R)))
23		breq1 2065	. . . . . . . . 9 ⊢ (y = w → (yRz ↔ wRz))
24	23	birexdv 1220	. . . . . . . 8 ⊢ (y = w → (∃z ∈ B yRz ↔ ∃z ∈ B wRz))
25	22, 24	imbi12d 474	. . . . . . 7 ⊢ (y = w → ((yRsup(B, A, R) → ∃z ∈ B yRz) ↔ (wRsup(B, A, R) → ∃z ∈ B wRz)))
26	25	rcla4v 1402	. . . . . 6 ⊢ (∀y ∈ A (yRsup(B, A, R) → ∃z ∈ B yRz) → (w ∈ A → (wRsup(B, A, R) → ∃z ∈ B wRz)))
27	21, 26	syl 12	. . . . 5 ⊢ (∃x ∈ A (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)) → (w ∈ A → (wRsup(B, A, R) → ∃z ∈ B wRz)))
28	27	com12 13	. . . 4 ⊢ (w ∈ A → (∃x ∈ A (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)) → (wRsup(B, A, R) → ∃z ∈ B wRz)))
29	5, 28	vtoclga 1387	. . 3 ⊢ (C ∈ A → (∃x ∈ A (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)) → (CRsup(B, A, R) → ∃z ∈ B CRz)))
30	29	com12 13	. 2 ⊢ (∃x ∈ A (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)) → (C ∈ A → (CRsup(B, A, R) → ∃z ∈ B CRz)))
31	30	imp3a 279	1 ⊢ (∃x ∈ A (∀y ∈ B ¬ xRy ∧ ∀y ∈ A (yRx → ∃z ∈ B yRz)) → ((C ∈ A ∧ CRsup(B, A, R)) → ∃z ∈ B CRz))

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:  supnub 2162  suplubi 2166  suprlub 4514

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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