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Mirrors > Home > NFE Home > Th. List > ssiinf | Unicode version |
Description: Subset theorem for an indexed intersection. (Contributed by FL, 15-Oct-2012.) (Proof shortened by Mario Carneiro, 14-Oct-2016.) |
Ref | Expression |
---|---|
ssiinf.1 |
Ref | Expression |
---|---|
ssiinf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2862 | . . . . 5 | |
2 | eliin 3974 | . . . . 5 | |
3 | 1, 2 | ax-mp 8 | . . . 4 |
4 | 3 | ralbii 2638 | . . 3 |
5 | ssiinf.1 | . . . 4 | |
6 | nfcv 2489 | . . . 4 | |
7 | 5, 6 | ralcomf 2769 | . . 3 |
8 | 4, 7 | bitri 240 | . 2 |
9 | dfss3 3263 | . 2 | |
10 | dfss3 3263 | . . 3 | |
11 | 10 | ralbii 2638 | . 2 |
12 | 8, 9, 11 | 3bitr4i 268 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wb 176 wcel 1710 wnfc 2476 wral 2614 cvv 2859 wss 3257 ciin 3970 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ral 2619 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-ss 3259 df-iin 3972 |
This theorem is referenced by: ssiin 4016 dmiin 4965 |
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