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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 5510 | . 2 ⊢ ( E We 𝐴 → E Or 𝐴) | |
2 | solin 5462 | . . 3 ⊢ (( E Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥)) | |
3 | epel 5433 | . . . 4 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦) | |
4 | biid 264 | . . . 4 ⊢ (𝑥 = 𝑦 ↔ 𝑥 = 𝑦) | |
5 | epel 5433 | . . . 4 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥) | |
6 | 3, 4, 5 | 3orbi123i 1153 | . . 3 ⊢ ((𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)) |
7 | 2, 6 | sylib 221 | . 2 ⊢ (( E Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)) |
8 | 1, 7 | sylan 583 | 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 399 ∨ w3o 1083 ∈ wcel 2111 class class class wbr 5030 E cep 5429 Or wor 5437 We wwe 5477 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-eprel 5430 df-so 5439 df-we 5480 |
This theorem is referenced by: tz7.7 6185 |
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