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Mirrors > Home > ILE Home > Th. List > uniss2 | Unicode version |
Description: A subclass condition on the members of two classes that implies a subclass relation on their unions. Proposition 8.6 of [TakeutiZaring] p. 59. (Contributed by NM, 22-Mar-2004.) |
Ref | Expression |
---|---|
uniss2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssuni 3602 | . . . . 5 | |
2 | 1 | expcom 109 | . . . 4 |
3 | 2 | rexlimiv 2427 | . . 3 |
4 | 3 | ralimi 2384 | . 2 |
5 | unissb 3610 | . 2 | |
6 | 4, 5 | sylibr 137 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wcel 1393 wral 2306 wrex 2307 wss 2917 cuni 3580 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-in 2924 df-ss 2931 df-uni 3581 |
This theorem is referenced by: unidif 3612 |
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