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Theorem oawordeu 7804
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) ∧ 𝐴𝐵) → ∃!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.
StepHypRef Expression
1 sseq1 3767 . . . 4 (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (𝐴𝐵 ↔ if(𝐴 ∈ On, 𝐴, ∅) ⊆ 𝐵))
2 oveq1 6820 . . . . . 6 (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (𝐴 +𝑜 𝑥) = (if(𝐴 ∈ On, 𝐴, ∅) +𝑜 𝑥))
32eqeq1d 2762 . . . . 5 (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → ((𝐴 +𝑜 𝑥) = 𝐵 ↔ (if(𝐴 ∈ On, 𝐴, ∅) +𝑜 𝑥) = 𝐵))
43reubidv 3265 . . . 4 (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (∃!𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵 ↔ ∃!𝑥 ∈ On (if(𝐴 ∈ On, 𝐴, ∅) +𝑜 𝑥) = 𝐵))
51, 4imbi12d 333 . . 3 (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → ((𝐴𝐵 → ∃!𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵) ↔ (if(𝐴 ∈ On, 𝐴, ∅) ⊆ 𝐵 → ∃!𝑥 ∈ On (if(𝐴 ∈ On, 𝐴, ∅) +𝑜 𝑥) = 𝐵)))
6 sseq2 3768 . . . 4 (𝐵 = if(𝐵 ∈ On, 𝐵, ∅) → (if(𝐴 ∈ On, 𝐴, ∅) ⊆ 𝐵 ↔ if(𝐴 ∈ On, 𝐴, ∅) ⊆ if(𝐵 ∈ On, 𝐵, ∅)))
7 eqeq2 2771 . . . . 5 (𝐵 = if(𝐵 ∈ On, 𝐵, ∅) → ((if(𝐴 ∈ On, 𝐴, ∅) +𝑜 𝑥) = 𝐵 ↔ (if(𝐴 ∈ On, 𝐴, ∅) +𝑜 𝑥) = if(𝐵 ∈ On, 𝐵, ∅)))
87reubidv 3265 . . . 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, ∅)))
96, 8imbi12d 333 . . 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 5939 . . . . 5 ∅ ∈ On
1110elimel 4294 . . . 4 if(𝐴 ∈ On, 𝐴, ∅) ∈ On
1210elimel 4294 . . . 4 if(𝐵 ∈ On, 𝐵, ∅) ∈ On
13 eqid 2760 . . . 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)}
1411, 12, 13oawordeulem 7803 . . 3 (if(𝐴 ∈ On, 𝐴, ∅) ⊆ if(𝐵 ∈ On, 𝐵, ∅) → ∃!𝑥 ∈ On (if(𝐴 ∈ On, 𝐴, ∅) +𝑜 𝑥) = if(𝐵 ∈ On, 𝐵, ∅))
155, 9, 14dedth2h 4284 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → ∃!𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵))
1615imp 444 1 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) → ∃!𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  ∃!wreu 3052  {crab 3054  wss 3715  c0 4058  ifcif 4230  Oncon0 5884  (class class class)co 6813   +𝑜 coa 7726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-oadd 7733
This theorem is referenced by:  oawordex  7806  oaf1o  7812  oaabs  7893  oaabs2  7894  finxpreclem4  33542
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