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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 5571 | . 2 ⊢ ( E We 𝐴 → E Or 𝐴) | |
2 | solin 5519 | . . 3 ⊢ (( E Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥)) | |
3 | epel 5489 | . . . 4 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦) | |
4 | biid 260 | . . . 4 ⊢ (𝑥 = 𝑦 ↔ 𝑥 = 𝑦) | |
5 | epel 5489 | . . . 4 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥) | |
6 | 3, 4, 5 | 3orbi123i 1154 | . . 3 ⊢ ((𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)) |
7 | 2, 6 | sylib 217 | . 2 ⊢ (( E Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)) |
8 | 1, 7 | sylan 579 | 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 1084 ∈ wcel 2108 class class class wbr 5070 E cep 5485 Or wor 5493 We wwe 5534 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pr 5347 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2817 df-ne 2943 df-ral 3068 df-rab 3072 df-v 3424 df-dif 3886 df-un 3888 df-nul 4254 df-if 4457 df-sn 4559 df-pr 4561 df-op 4565 df-br 5071 df-opab 5133 df-eprel 5486 df-so 5495 df-we 5537 |
This theorem is referenced by: tz7.7 6277 |
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