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

Theorem oawordex 7524
Description: Existence theorem for weak ordering of ordinal sum. Proposition 8.8 of [TakeutiZaring] p. 59 and its converse. See oawordeu 7522 for uniqueness. (Contributed by NM, 12-Dec-2004.)
Assertion
Ref Expression
oawordex ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ ∃𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem oawordex
StepHypRef Expression
1 oawordeu 7522 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴𝐵) → ∃!𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵)
21ex 449 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → ∃!𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵))
3 reurex 3137 . . 3 (∃!𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵 → ∃𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵)
42, 3syl6 34 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 → ∃𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵))
5 oawordexr 7523 . . . 4 ((𝐴 ∈ On ∧ ∃𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵) → 𝐴𝐵)
65ex 449 . . 3 (𝐴 ∈ On → (∃𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵𝐴𝐵))
76adantr 480 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵𝐴𝐵))
84, 7impbid 201 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ ∃𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wrex 2897  ∃!wreu 2898  wss 3540  Oncon0 5640  (class class class)co 6549   +𝑜 coa 7444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451
This theorem is referenced by:  oaordex  7525  oaass  7528  odi  7546  omeulem1  7549
  Copyright terms: Public domain W3C validator