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Mirrors > Home > ILE Home > Th. List > trintssm | Unicode version |
Description: Any inhabited transitive class includes its intersection. Similar to Exercise 3 in [TakeutiZaring] p. 44 (which mistakenly does not include the inhabitedness hypothesis). (Contributed by Jim Kingdon, 22-Aug-2018.) |
Ref | Expression |
---|---|
trintssm |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | intss1 3844 | . . . 4 | |
2 | trss 4094 | . . . . 5 | |
3 | 2 | com12 30 | . . . 4 |
4 | sstr2 3154 | . . . 4 | |
5 | 1, 3, 4 | sylsyld 58 | . . 3 |
6 | 5 | exlimiv 1591 | . 2 |
7 | 6 | impcom 124 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wex 1485 wcel 2141 wss 3121 cint 3829 wtr 4085 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-v 2732 df-in 3127 df-ss 3134 df-uni 3795 df-int 3830 df-tr 4086 |
This theorem is referenced by: (None) |
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