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

Theorem epweonALT 7795
Description: Alternate proof of epweon 7794, shorter but requiring ax-un 7754. (Contributed by NM, 1-Nov-2003.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
epweonALT E We On

Proof of Theorem epweonALT
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 onfr 6425 . 2 E Fr On
2 eloni 6396 . . . 4 (𝑥 ∈ On → Ord 𝑥)
3 eloni 6396 . . . 4 (𝑦 ∈ On → Ord 𝑦)
4 ordtri3or 6418 . . . . 5 ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
5 epel 5592 . . . . . 6 (𝑥 E 𝑦𝑥𝑦)
6 biid 261 . . . . . 6 (𝑥 = 𝑦𝑥 = 𝑦)
7 epel 5592 . . . . . 6 (𝑦 E 𝑥𝑦𝑥)
85, 6, 73orbi123i 1155 . . . . 5 ((𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥) ↔ (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
94, 8sylibr 234 . . . 4 ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥))
102, 3, 9syl2an 596 . . 3 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥))
1110rgen2 3197 . 2 𝑥 ∈ On ∀𝑦 ∈ On (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥)
12 dfwe2 7793 . 2 ( E We On ↔ ( E Fr On ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥)))
131, 11, 12mpbir2an 711 1 E We On
Colors of variables: wff setvar class
Syntax hints:  wa 395  w3o 1085  wcel 2106  wral 3059   class class class wbr 5148   E cep 5588   Fr wfr 5638   We wwe 5640  Ord word 6385  Oncon0 6386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator