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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 3580 | . . . 4 ⊢ (A = B → ∪ A = ∪ B) | |
2 | 1 | sseq1d 2966 | . . 3 ⊢ (A = B → (∪ A ⊆ A ↔ ∪ B ⊆ A)) |
3 | sseq2 2961 | . . 3 ⊢ (A = B → (∪ B ⊆ A ↔ ∪ B ⊆ B)) | |
4 | 2, 3 | bitrd 177 | . 2 ⊢ (A = B → (∪ A ⊆ A ↔ ∪ B ⊆ B)) |
5 | df-tr 3846 | . 2 ⊢ (Tr A ↔ ∪ A ⊆ A) | |
6 | df-tr 3846 | . 2 ⊢ (Tr B ↔ ∪ B ⊆ B) | |
7 | 4, 5, 6 | 3bitr4g 212 | 1 ⊢ (A = B → (Tr A ↔ Tr B)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 = wceq 1242 ⊆ wss 2911 ∪ cuni 3571 Tr wtr 3845 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-rex 2306 df-in 2918 df-ss 2925 df-uni 3572 df-tr 3846 |
This theorem is referenced by: truni 3859 ordeq 4075 ordsucim 4192 ordom 4272 |
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