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| Description: Alternate proof of epweon 7796, shorter but requiring ax-un 7756. (Contributed by NM, 1-Nov-2003.) (Proof modification is discouraged.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| epweonALT | ⊢ E We On | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | onfr 6422 | . 2 ⊢ E Fr On | |
| 2 | eloni 6393 | . . . 4 ⊢ (𝑥 ∈ On → Ord 𝑥) | |
| 3 | eloni 6393 | . . . 4 ⊢ (𝑦 ∈ On → Ord 𝑦) | |
| 4 | ordtri3or 6415 | . . . . 5 ⊢ ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)) | |
| 5 | epel 5586 | . . . . . 6 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦) | |
| 6 | biid 261 | . . . . . 6 ⊢ (𝑥 = 𝑦 ↔ 𝑥 = 𝑦) | |
| 7 | epel 5586 | . . . . . 6 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥) | |
| 8 | 5, 6, 7 | 3orbi123i 1156 | . . . . 5 ⊢ ((𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)) | 
| 9 | 4, 8 | sylibr 234 | . . . 4 ⊢ ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥)) | 
| 10 | 2, 3, 9 | syl2an 596 | . . 3 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥)) | 
| 11 | 10 | rgen2 3198 | . 2 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥) | 
| 12 | dfwe2 7795 | . 2 ⊢ ( E We On ↔ ( E Fr On ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥))) | |
| 13 | 1, 11, 12 | mpbir2an 711 | 1 ⊢ E We On | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 ∨ w3o 1085 ∈ wcel 2107 ∀wral 3060 class class class wbr 5142 E cep 5582 Fr wfr 5633 We wwe 5635 Ord word 6382 Oncon0 6383 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-tr 5259 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-ord 6386 df-on 6387 | 
| This theorem is referenced by: (None) | 
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