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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 4923 | . . . 4 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵) | |
2 | 1 | sseq1d 4027 | . . 3 ⊢ (𝐴 = 𝐵 → (∪ 𝐴 ⊆ 𝐴 ↔ ∪ 𝐵 ⊆ 𝐴)) |
3 | sseq2 4022 | . . 3 ⊢ (𝐴 = 𝐵 → (∪ 𝐵 ⊆ 𝐴 ↔ ∪ 𝐵 ⊆ 𝐵)) | |
4 | 2, 3 | bitrd 279 | . 2 ⊢ (𝐴 = 𝐵 → (∪ 𝐴 ⊆ 𝐴 ↔ ∪ 𝐵 ⊆ 𝐵)) |
5 | df-tr 5266 | . 2 ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴) | |
6 | df-tr 5266 | . 2 ⊢ (Tr 𝐵 ↔ ∪ 𝐵 ⊆ 𝐵) | |
7 | 4, 5, 6 | 3bitr4g 314 | 1 ⊢ (𝐴 = 𝐵 → (Tr 𝐴 ↔ Tr 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ⊆ wss 3963 ∪ cuni 4912 Tr wtr 5265 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-ss 3980 df-uni 4913 df-tr 5266 |
This theorem is referenced by: truni 5281 trint 5283 ordeq 6393 trcl 9766 tz9.1 9767 tz9.1c 9768 tctr 9778 tcmin 9779 tc2 9780 r1tr 9814 r1elssi 9843 tcrank 9922 iswun 10742 tskr1om2 10806 elgrug 10830 grutsk 10860 dfon2lem1 35765 dfon2lem3 35767 dfon2lem4 35768 dfon2lem5 35769 dfon2lem6 35770 dfon2lem7 35771 dfon2lem8 35772 dfon2 35774 dford3lem1 43015 dford3lem2 43016 nadd1rabtr 43378 wfaxext 44949 wfaxrep 44950 |
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