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| Mirrors > Home > MPE Home > Th. List > trss | Structured version Visualization version GIF version | ||
| Description: An element of a transitive class is a subset of the class. (Contributed by NM, 7-Aug-1994.) (Proof shortened by JJ, 26-Jul-2021.) |
| Ref | Expression |
|---|---|
| trss | ⊢ (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dftr3 5265 | . 2 ⊢ (Tr 𝐴 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴) | |
| 2 | sseq1 4009 | . . 3 ⊢ (𝑥 = 𝐵 → (𝑥 ⊆ 𝐴 ↔ 𝐵 ⊆ 𝐴)) | |
| 3 | 2 | rspccv 3619 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)) |
| 4 | 1, 3 | sylbi 217 | 1 ⊢ (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ∀wral 3061 ⊆ wss 3951 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-ral 3062 df-v 3482 df-ss 3968 df-uni 4908 df-tr 5260 |
| This theorem is referenced by: trin 5271 triun 5274 triin 5276 trintss 5278 tz7.2 5668 trpred 6352 ordelss 6400 ordelord 6406 tz7.7 6410 trsucss 6472 tc2 9782 tcel 9785 r1ord3g 9819 r1ord2 9821 r1pwss 9824 rankwflemb 9833 r1elwf 9836 r1elssi 9845 uniwf 9859 itunitc1 10460 wunelss 10748 tskr1om2 10808 tskuni 10823 tskurn 10829 gruelss 10834 dfon2lem6 35789 dfon2lem9 35792 setindtr 43036 dford3lem1 43038 ordelordALT 44557 trsspwALT 44838 trsspwALT2 44839 trsspwALT3 44840 pwtrVD 44844 ordelordALTVD 44887 ralabso 44985 rexabso 44986 modelaxrep 44998 omelaxinf2 45006 |
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