Theorem oawordeu 3157

Description: Existence theorem for weak ordering of ordinal sum. Proposition 8.8 of [TakeutiZaring] p. 59.

Assertion

Ref	Expression
oawordeu	⊢ (((A ∈ On ∧ B ∈ On) ∧ A ⊆ B) → ∃!x ∈ On (A +o x) = B)

Distinct variable group(s):   x,A   x,B

Proof of Theorem oawordeu

Step	Hyp	Ref	Expression
1		sseq1 1521	. . . 4 ⊢ (A = if(A ∈ On, A, ∅) → (A ⊆ B ↔ if(A ∈ On, A, ∅) ⊆ B))
2		opreq1 3006	. . . . . 6 ⊢ (A = if(A ∈ On, A, ∅) → (A +o x) = (if(A ∈ On, A, ∅) +o x))
3	2	cleq1d 1109	. . . . 5 ⊢ (A = if(A ∈ On, A, ∅) → ((A +o x) = B ↔ (if(A ∈ On, A, ∅) +o x) = B))
4	3	bireudv 1318	. . . 4 ⊢ (A = if(A ∈ On, A, ∅) → (∃!x ∈ On (A +o x) = B ↔ ∃!x ∈ On (if(A ∈ On, A, ∅) +o x) = B))
5	1, 4	imbi12d 474	. . 3 ⊢ (A = if(A ∈ On, A, ∅) → ((A ⊆ B → ∃!x ∈ On (A +o x) = B) ↔ (if(A ∈ On, A, ∅) ⊆ B → ∃!x ∈ On (if(A ∈ On, A, ∅) +o x) = B)))
6		sseq2 1522	. . . 4 ⊢ (B = if(B ∈ On, B, ∅) → (if(A ∈ On, A, ∅) ⊆ B ↔ if(A ∈ On, A, ∅) ⊆ if(B ∈ On, B, ∅)))
7		cleq2 1110	. . . . 5 ⊢ (B = if(B ∈ On, B, ∅) → ((if(A ∈ On, A, ∅) +o x) = B ↔ (if(A ∈ On, A, ∅) +o x) = if(B ∈ On, B, ∅)))
8	7	bireudv 1318	. . . 4 ⊢ (B = if(B ∈ On, B, ∅) → (∃!x ∈ On (if(A ∈ On, A, ∅) +o x) = B ↔ ∃!x ∈ On (if(A ∈ On, A, ∅) +o x) = if(B ∈ On, B, ∅)))
9	6, 8	imbi12d 474	. . 3 ⊢ (B = if(B ∈ On, B, ∅) → ((if(A ∈ On, A, ∅) ⊆ B → ∃!x ∈ On (if(A ∈ On, A, ∅) +o x) = B) ↔ (if(A ∈ On, A, ∅) ⊆ if(B ∈ On, B, ∅) → ∃!x ∈ On (if(A ∈ On, A, ∅) +o x) = if(B ∈ On, B, ∅))))
10		0elon 2277	. . . . 5 ⊢ ∅ ∈ On
11	10	elimel 1793	. . . 4 ⊢ if(A ∈ On, A, ∅) ∈ On
12	10	elimel 1793	. . . 4 ⊢ if(B ∈ On, B, ∅) ∈ On
13		cleqid 1102	. . . 4 ⊢ {y ∈ On∣if(B ∈ On, B, ∅) ⊆ (if(A ∈ On, A, ∅) +o y)} = {y ∈ On∣if(B ∈ On, B, ∅) ⊆ (if(A ∈ On, A, ∅) +o y)}
14	11, 12, 13	oawordeulem 3156	. . 3 ⊢ (if(A ∈ On, A, ∅) ⊆ if(B ∈ On, B, ∅) → ∃!x ∈ On (if(A ∈ On, A, ∅) +o x) = if(B ∈ On, B, ∅))
15	5, 9, 14	dedth2h 1787	. 2 ⊢ ((A ∈ On ∧ B ∈ On) → (A ⊆ B → ∃!x ∈ On (A +o x) = B))
16	15	imp 277	1 ⊢ (((A ∈ On ∧ B ∈ On) ∧ A ⊆ B) → ∃!x ∈ On (A +o x) = B)

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∃!wreu 1203  {crab 1204   ⊆ wss 1487  ∅c0 1707  ifcif 1776  Oncon0 2199  (class class class)co 3001   +_o coa 3101

This theorem is referenced by:  oawordex 3159

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-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-oadd 3106

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