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| Mirrors > Home > MPE Home > Th. List > wecmpep | Structured version Visualization version GIF version | ||
| Description: The elements of a class well-ordered by membership are comparable. (Contributed by NM, 17-May-1994.) |
| Ref | Expression |
|---|---|
| wecmpep | ⊢ (( E We 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | weso 5676 | . 2 ⊢ ( E We 𝐴 → E Or 𝐴) | |
| 2 | solin 5619 | . . 3 ⊢ (( E Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥)) | |
| 3 | epel 5587 | . . . 4 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦) | |
| 4 | biid 261 | . . . 4 ⊢ (𝑥 = 𝑦 ↔ 𝑥 = 𝑦) | |
| 5 | epel 5587 | . . . 4 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥) | |
| 6 | 3, 4, 5 | 3orbi123i 1157 | . . 3 ⊢ ((𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)) |
| 7 | 2, 6 | sylib 218 | . 2 ⊢ (( E Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)) |
| 8 | 1, 7 | sylan 580 | 1 ⊢ (( E We 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∨ w3o 1086 ∈ wcel 2108 class class class wbr 5143 E cep 5583 Or wor 5591 We wwe 5636 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-eprel 5584 df-so 5593 df-we 5639 |
| This theorem is referenced by: tz7.7 6410 |
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