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Mirrors > Home > MPE Home > Th. List > wetrep | Structured version Visualization version GIF version |
Description: On a class well-ordered by membership, the membership predicate is transitive. (Contributed by NM, 22-Apr-1994.) |
Ref | Expression |
---|---|
wetrep | ⊢ (( E We 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | weso 5669 | . . 3 ⊢ ( E We 𝐴 → E Or 𝐴) | |
2 | sotr 5614 | . . 3 ⊢ (( E Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧)) | |
3 | 1, 2 | sylan 578 | . 2 ⊢ (( E We 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧)) |
4 | epel 5585 | . . 3 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦) | |
5 | epel 5585 | . . 3 ⊢ (𝑦 E 𝑧 ↔ 𝑦 ∈ 𝑧) | |
6 | 4, 5 | anbi12i 626 | . 2 ⊢ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) ↔ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧)) |
7 | epel 5585 | . 2 ⊢ (𝑥 E 𝑧 ↔ 𝑥 ∈ 𝑧) | |
8 | 3, 6, 7 | 3imtr3g 294 | 1 ⊢ (( E We 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 ∈ wcel 2098 class class class wbr 5149 E cep 5581 Or wor 5589 We wwe 5632 |
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 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2930 df-ral 3051 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5150 df-opab 5212 df-eprel 5582 df-po 5590 df-so 5591 df-we 5635 |
This theorem is referenced by: wefrc 5672 ordelord 6393 |
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