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Theorem epweonALT 7759
Description: Alternate proof of epweon 7758, shorter but requiring ax-un 7721. (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 6400 . 2 E Fr On
2 eloni 6371 . . . 4 (𝑥 ∈ On → Ord 𝑥)
3 eloni 6371 . . . 4 (𝑦 ∈ On → Ord 𝑦)
4 ordtri3or 6393 . . . . 5 ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
5 epel 5582 . . . . . 6 (𝑥 E 𝑦𝑥𝑦)
6 biid 260 . . . . . 6 (𝑥 = 𝑦𝑥 = 𝑦)
7 epel 5582 . . . . . 6 (𝑦 E 𝑥𝑦𝑥)
85, 6, 73orbi123i 1156 . . . . 5 ((𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥) ↔ (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
94, 8sylibr 233 . . . 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 7757 . 2 ( E We On ↔ ( E Fr On ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥)))
131, 11, 12mpbir2an 709 1 E We On
Colors of variables: wff setvar class
Syntax hints:  wa 396  w3o 1086  wcel 2106  wral 3061   class class class wbr 5147   E cep 5578   Fr wfr 5627   We wwe 5629  Ord word 6360  Oncon0 6361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-ord 6364  df-on 6365
This theorem is referenced by: (None)
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