MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oawordeu Structured version   Visualization version   GIF version

Theorem oawordeu 8593
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 4009 . . . 4 (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (𝐴𝐵 ↔ if(𝐴 ∈ On, 𝐴, ∅) ⊆ 𝐵))
2 oveq1 7438 . . . . . 6 (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (𝐴 +o 𝑥) = (if(𝐴 ∈ On, 𝐴, ∅) +o 𝑥))
32eqeq1d 2739 . . . . 5 (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → ((𝐴 +o 𝑥) = 𝐵 ↔ (if(𝐴 ∈ On, 𝐴, ∅) +o 𝑥) = 𝐵))
43reubidv 3398 . . . 4 (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (∃!𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵 ↔ ∃!𝑥 ∈ On (if(𝐴 ∈ On, 𝐴, ∅) +o 𝑥) = 𝐵))
51, 4imbi12d 344 . . 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 4010 . . . 4 (𝐵 = if(𝐵 ∈ On, 𝐵, ∅) → (if(𝐴 ∈ On, 𝐴, ∅) ⊆ 𝐵 ↔ if(𝐴 ∈ On, 𝐴, ∅) ⊆ if(𝐵 ∈ On, 𝐵, ∅)))
7 eqeq2 2749 . . . . 5 (𝐵 = if(𝐵 ∈ On, 𝐵, ∅) → ((if(𝐴 ∈ On, 𝐴, ∅) +o 𝑥) = 𝐵 ↔ (if(𝐴 ∈ On, 𝐴, ∅) +o 𝑥) = if(𝐵 ∈ On, 𝐵, ∅)))
87reubidv 3398 . . . 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 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 6438 . . . . 5 ∅ ∈ On
1110elimel 4595 . . . 4 if(𝐴 ∈ On, 𝐴, ∅) ∈ On
1210elimel 4595 . . . 4 if(𝐵 ∈ On, 𝐵, ∅) ∈ On
13 eqid 2737 . . . 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 8592 . . 3 (if(𝐴 ∈ On, 𝐴, ∅) ⊆ if(𝐵 ∈ On, 𝐵, ∅) → ∃!𝑥 ∈ On (if(𝐴 ∈ On, 𝐴, ∅) +o 𝑥) = if(𝐵 ∈ On, 𝐵, ∅))
155, 9, 14dedth2h 4585 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → ∃!𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵))
1615imp 406 1 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) → ∃!𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  ∃!wreu 3378  {crab 3436  wss 3951  c0 4333  ifcif 4525  Oncon0 6384  (class class class)co 7431   +o coa 8503
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 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-oadd 8510
This theorem is referenced by:  oawordex  8595  oaf1o  8601  oaabs  8686  oaabs2  8687  finxpreclem4  37395  tfsconcatlem  43349  tfsconcatfv  43354
  Copyright terms: Public domain W3C validator