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Mirrors > Home > ILE Home > Th. List > treq | GIF version |
Description: Equality theorem for the transitive class predicate. (Contributed by NM, 17-Sep-1993.) |
Ref | Expression |
---|---|
treq | ⊢ (𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unieq 3610 | . . . 4 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵) | |
2 | 1 | sseq1d 3026 | . . 3 ⊢ (𝐴 = 𝐵 → (∪ 𝐴 ⊆ 𝐴 ↔ ∪ 𝐵 ⊆ 𝐴)) |
3 | sseq2 3021 | . . 3 ⊢ (𝐴 = 𝐵 → (∪ 𝐵 ⊆ 𝐴 ↔ ∪ 𝐵 ⊆ 𝐵)) | |
4 | 2, 3 | bitrd 186 | . 2 ⊢ (𝐴 = 𝐵 → (∪ 𝐴 ⊆ 𝐴 ↔ ∪ 𝐵 ⊆ 𝐵)) |
5 | df-tr 3876 | . 2 ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴) | |
6 | df-tr 3876 | . 2 ⊢ (Tr 𝐵 ↔ ∪ 𝐵 ⊆ 𝐵) | |
7 | 4, 5, 6 | 3bitr4g 221 | 1 ⊢ (𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 = wceq 1284 ⊆ wss 2973 ∪ cuni 3601 Tr wtr 3875 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-rex 2354 df-in 2979 df-ss 2986 df-uni 3602 df-tr 3876 |
This theorem is referenced by: truni 3889 ordeq 4127 ordsucim 4244 ordom 4347 |
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