Theorem oawordeu 8386

Description: Existence theorem for weak ordering of ordinal sum. Proposition 8.8 of [TakeutiZaring] p. 59. (Contributed by NM, 11-Dec-2004.)

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem oawordeu

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseq1 3946	. . . 4 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (𝐴 ⊆ 𝐵 ↔ if(𝐴 ∈ On, 𝐴, ∅) ⊆ 𝐵))
2		oveq1 7282	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (𝐴 +o 𝑥) = (if(𝐴 ∈ On, 𝐴, ∅) +o 𝑥))
3	2	eqeq1d 2740	. . . . 5 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → ((𝐴 +o 𝑥) = 𝐵 ↔ (if(𝐴 ∈ On, 𝐴, ∅) +o 𝑥) = 𝐵))
4	3	reubidv 3323	. . . 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 345	. . 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 3947	. . . 4 ⊢ (𝐵 = if(𝐵 ∈ On, 𝐵, ∅) → (if(𝐴 ∈ On, 𝐴, ∅) ⊆ 𝐵 ↔ if(𝐴 ∈ On, 𝐴, ∅) ⊆ if(𝐵 ∈ On, 𝐵, ∅)))
7		eqeq2 2750	. . . . 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	reubidv 3323	. . . 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 345	. . 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 6319	. . . . 5 ⊢ ∅ ∈ On
11	10	elimel 4528	. . . 4 ⊢ if(𝐴 ∈ On, 𝐴, ∅) ∈ On
12	10	elimel 4528	. . . 4 ⊢ if(𝐵 ∈ On, 𝐵, ∅) ∈ On
13		eqid 2738	. . . 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 8385	. . 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 4518	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → ∃!𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵))
16	15	imp 407	1 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ∃!𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∃!wreu 3066  {crab 3068   ⊆ wss 3887  ∅c0 4256  ifcif 4459  Oncon0 6266  (class class class)co 7275   +_o coa 8294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-oadd 8301

This theorem is referenced by:  oawordex  8388  oaf1o  8394  oaabs  8478  oaabs2  8479  finxpreclem4  35565