| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 5676 | . . 3 ⊢ ( E We 𝐴 → E Or 𝐴) | |
| 2 | sotr 5617 | . . 3 ⊢ (( E Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧)) | |
| 3 | 1, 2 | sylan 580 | . 2 ⊢ (( E We 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧)) |
| 4 | epel 5587 | . . 3 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦) | |
| 5 | epel 5587 | . . 3 ⊢ (𝑦 E 𝑧 ↔ 𝑦 ∈ 𝑧) | |
| 6 | 4, 5 | anbi12i 628 | . 2 ⊢ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) ↔ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧)) |
| 7 | epel 5587 | . 2 ⊢ (𝑥 E 𝑧 ↔ 𝑥 ∈ 𝑧) | |
| 8 | 3, 6, 7 | 3imtr3g 295 | 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 395 ∧ w3a 1087 ∈ 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-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-po 5592 df-so 5593 df-we 5639 |
| This theorem is referenced by: wefrc 5679 ordelord 6406 |
| Copyright terms: Public domain | W3C validator |