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Theorem epweonALT 7776
Description: Alternate proof of epweon 7775, shorter but requiring ax-un 7738. (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 6403 . 2 E Fr On
2 eloni 6374 . . . 4 (𝑥 ∈ On → Ord 𝑥)
3 eloni 6374 . . . 4 (𝑦 ∈ On → Ord 𝑦)
4 ordtri3or 6396 . . . . 5 ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
5 epel 5579 . . . . . 6 (𝑥 E 𝑦𝑥𝑦)
6 biid 260 . . . . . 6 (𝑥 = 𝑦𝑥 = 𝑦)
7 epel 5579 . . . . . 6 (𝑦 E 𝑥𝑦𝑥)
85, 6, 73orbi123i 1153 . . . . 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 594 . . 3 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥))
1110rgen2 3188 . 2 𝑥 ∈ On ∀𝑦 ∈ On (𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥)
12 dfwe2 7774 . 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 394  w3o 1083  wcel 2098  wral 3051   class class class wbr 5143   E cep 5575   Fr wfr 5624   We wwe 5626  Ord word 6363  Oncon0 6364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-tr 5261  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-ord 6367  df-on 6368
This theorem is referenced by: (None)
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