Theorem oawordeu 8565

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 3984	. . . 4 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (𝐴 ⊆ 𝐵 ↔ if(𝐴 ∈ On, 𝐴, ∅) ⊆ 𝐵))
2		oveq1 7410	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (𝐴 +o 𝑥) = (if(𝐴 ∈ On, 𝐴, ∅) +o 𝑥))
3	2	eqeq1d 2737	. . . . 5 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → ((𝐴 +o 𝑥) = 𝐵 ↔ (if(𝐴 ∈ On, 𝐴, ∅) +o 𝑥) = 𝐵))
4	3	reubidv 3377	. . . 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 344	. . 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 3985	. . . 4 ⊢ (𝐵 = if(𝐵 ∈ On, 𝐵, ∅) → (if(𝐴 ∈ On, 𝐴, ∅) ⊆ 𝐵 ↔ if(𝐴 ∈ On, 𝐴, ∅) ⊆ if(𝐵 ∈ On, 𝐵, ∅)))
7		eqeq2 2747	. . . . 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 3377	. . . 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 344	. . 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 6407	. . . . 5 ⊢ ∅ ∈ On
11	10	elimel 4570	. . . 4 ⊢ if(𝐴 ∈ On, 𝐴, ∅) ∈ On
12	10	elimel 4570	. . . 4 ⊢ if(𝐵 ∈ On, 𝐵, ∅) ∈ On
13		eqid 2735	. . . 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 8564	. . 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 4560	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → ∃!𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵))
16	15	imp 406	1 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ∃!𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∃!wreu 3357  {crab 3415   ⊆ wss 3926  ∅c0 4308  ifcif 4500  Oncon0 6352  (class class class)co 7403   +_o coa 8475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7727

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-ov 7406  df-oprab 7407  df-mpo 7408  df-om 7860  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-oadd 8482

This theorem is referenced by:  oawordex  8567  oaf1o  8573  oaabs  8658  oaabs2  8659  finxpreclem4  37358  tfsconcatlem  43307  tfsconcatfv  43312