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Mirrors > Home > ILE Home > Th. List > iinss | Unicode version |
Description: Subset implication for an indexed intersection. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
iinss |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2733 | . . . 4 | |
2 | eliin 3876 | . . . 4 | |
3 | 1, 2 | ax-mp 5 | . . 3 |
4 | ssel 3141 | . . . . 5 | |
5 | 4 | reximi 2567 | . . . 4 |
6 | r19.36av 2621 | . . . 4 | |
7 | 5, 6 | syl 14 | . . 3 |
8 | 3, 7 | syl5bi 151 | . 2 |
9 | 8 | ssrdv 3153 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wcel 2141 wral 2448 wrex 2449 cvv 2730 wss 3121 ciin 3872 |
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-rex 2454 df-v 2732 df-in 3127 df-ss 3134 df-iin 3874 |
This theorem is referenced by: riinm 3943 reliin 4731 cnviinm 5150 iinerm 6581 |
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