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Mirrors > Home > NFE Home > Th. List > ssofss | Unicode version |
Description: Condition for subset when is already known to be a subset. (Contributed by SF, 13-Jan-2015.) |
Ref | Expression |
---|---|
ssofss |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2862 | . . . . . . . 8 | |
2 | 1 | elcompl 3225 | . . . . . . 7 ∼ |
3 | ssel 3267 | . . . . . . . 8 | |
4 | 3 | con3d 125 | . . . . . . 7 |
5 | 2, 4 | syl5bi 208 | . . . . . 6 ∼ |
6 | 5 | imp 418 | . . . . 5 ∼ |
7 | 6 | pm2.21d 98 | . . . 4 ∼ |
8 | 7 | ralrimiva 2697 | . . 3 ∼ |
9 | 8 | biantrud 493 | . 2 ∼ |
10 | ralv 2872 | . . . 4 | |
11 | uncompl 4074 | . . . . 5 ∼ | |
12 | 11 | raleqi 2811 | . . . 4 ∼ |
13 | dfss2 3262 | . . . 4 | |
14 | 10, 12, 13 | 3bitr4ri 269 | . . 3 ∼ |
15 | ralunb 3444 | . . 3 ∼ ∼ | |
16 | 14, 15 | bitri 240 | . 2 ∼ |
17 | 9, 16 | syl6rbbr 255 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 176 wa 358 wal 1540 wcel 1710 wral 2614 cvv 2859 ∼ ccompl 3205 cun 3207 wss 3257 |
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-un 3214 df-ss 3259 |
This theorem is referenced by: ssofeq 4077 ssrelk 4211 |
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