Theorem oawordeu 8484

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 3942	. . . 4 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (𝐴 ⊆ 𝐵 ↔ if(𝐴 ∈ On, 𝐴, ∅) ⊆ 𝐵))
2		oveq1 7367	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (𝐴 +o 𝑥) = (if(𝐴 ∈ On, 𝐴, ∅) +o 𝑥))
3	2	eqeq1d 2743	. . . . 5 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → ((𝐴 +o 𝑥) = 𝐵 ↔ (if(𝐴 ∈ On, 𝐴, ∅) +o 𝑥) = 𝐵))
4	3	reubidv 3362	. . . 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 346	. . 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 3943	. . . 4 ⊢ (𝐵 = if(𝐵 ∈ On, 𝐵, ∅) → (if(𝐴 ∈ On, 𝐴, ∅) ⊆ 𝐵 ↔ if(𝐴 ∈ On, 𝐴, ∅) ⊆ if(𝐵 ∈ On, 𝐵, ∅)))
7		eqeq2 2753	. . . . 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 3362	. . . 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 346	. . 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 6369	. . . . 5 ⊢ ∅ ∈ On
11	10	elimel 4527	. . . 4 ⊢ if(𝐴 ∈ On, 𝐴, ∅) ∈ On
12	10	elimel 4527	. . . 4 ⊢ if(𝐵 ∈ On, 𝐵, ∅) ∈ On
13		eqid 2741	. . . 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 8483	. . 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 4517	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → ∃!𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵))
16	15	imp 408	1 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ∃!𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548   ∈ wcel 2121  ∃!wreu 3344  {crab 3393   ⊆ wss 3885  ∅c0 4264  ifcif 4457  Oncon0 6314  (class class class)co 7360   +_o coa 8396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-int 4881  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-oadd 8403

This theorem is referenced by:  oawordex  8486  oaf1o  8492  oaabs  8578  oaabs2  8579  fineqvnttrclselem1  35317  fineqvnttrclselem2  35318  fineqvnttrclse  35320  finxpreclem4  37771  tfsconcatlem  43796  tfsconcatfv  43801