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| 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 4918 | . . . 4 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵) | |
| 2 | 1 | sseq1d 4015 | . . 3 ⊢ (𝐴 = 𝐵 → (∪ 𝐴 ⊆ 𝐴 ↔ ∪ 𝐵 ⊆ 𝐴)) | 
| 3 | sseq2 4010 | . . 3 ⊢ (𝐴 = 𝐵 → (∪ 𝐵 ⊆ 𝐴 ↔ ∪ 𝐵 ⊆ 𝐵)) | |
| 4 | 2, 3 | bitrd 279 | . 2 ⊢ (𝐴 = 𝐵 → (∪ 𝐴 ⊆ 𝐴 ↔ ∪ 𝐵 ⊆ 𝐵)) | 
| 5 | df-tr 5260 | . 2 ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴) | |
| 6 | df-tr 5260 | . 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 1540 ⊆ wss 3951 ∪ cuni 4907 Tr wtr 5259 | 
| 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 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-ss 3968 df-uni 4908 df-tr 5260 | 
| This theorem is referenced by: truni 5275 trint 5277 ordeq 6391 trcl 9768 tz9.1 9769 tz9.1c 9770 tctr 9780 tcmin 9781 tc2 9782 r1tr 9816 r1elssi 9845 tcrank 9924 iswun 10744 tskr1om2 10808 elgrug 10832 grutsk 10862 dfon2lem1 35784 dfon2lem3 35786 dfon2lem4 35787 dfon2lem5 35788 dfon2lem6 35789 dfon2lem7 35790 dfon2lem8 35791 dfon2 35793 dford3lem1 43038 dford3lem2 43039 nadd1rabtr 43401 wfaxext 45010 wfaxrep 45011 wfaxpow 45014 wfaxinf2 45018 wfac8prim 45019 | 
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