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Mirrors > Home > MPE Home > Th. List > treq | Structured version Visualization version 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 4380 | . . . 4 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵) | |
2 | 1 | sseq1d 3595 | . . 3 ⊢ (𝐴 = 𝐵 → (∪ 𝐴 ⊆ 𝐴 ↔ ∪ 𝐵 ⊆ 𝐴)) |
3 | sseq2 3590 | . . 3 ⊢ (𝐴 = 𝐵 → (∪ 𝐵 ⊆ 𝐴 ↔ ∪ 𝐵 ⊆ 𝐵)) | |
4 | 2, 3 | bitrd 267 | . 2 ⊢ (𝐴 = 𝐵 → (∪ 𝐴 ⊆ 𝐴 ↔ ∪ 𝐵 ⊆ 𝐵)) |
5 | df-tr 4681 | . 2 ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴) | |
6 | df-tr 4681 | . 2 ⊢ (Tr 𝐵 ↔ ∪ 𝐵 ⊆ 𝐵) | |
7 | 4, 5, 6 | 3bitr4g 302 | 1 ⊢ (𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ⊆ wss 3540 ∪ cuni 4372 Tr wtr 4680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-in 3547 df-ss 3554 df-uni 4373 df-tr 4681 |
This theorem is referenced by: truni 4695 ordeq 5647 trcl 8487 tz9.1 8488 tz9.1c 8489 tctr 8499 tcmin 8500 tc2 8501 r1tr 8522 r1elssi 8551 tcrank 8630 iswun 9405 tskr1om2 9469 elgrug 9493 grutsk 9523 dfon2lem1 30932 dfon2lem3 30934 dfon2lem4 30935 dfon2lem5 30936 dfon2lem6 30937 dfon2lem7 30938 dfon2lem8 30939 dfon2 30941 dford3lem1 36611 dford3lem2 36612 |
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