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Mirrors > Home > NFE Home > Th. List > sstp | Unicode version |
Description: The subsets of a triple. (Contributed by Mario Carneiro, 2-Jul-2016.) |
Ref | Expression |
---|---|
sstp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-tp 3744 | . . 3 | |
2 | 1 | sseq2i 3297 | . 2 |
3 | 0ss 3580 | . . 3 | |
4 | 3 | biantrur 492 | . 2 |
5 | ssunsn2 3866 | . . 3 | |
6 | 3 | biantrur 492 | . . . . 5 |
7 | sspr 3870 | . . . . 5 | |
8 | 6, 7 | bitr3i 242 | . . . 4 |
9 | uncom 3409 | . . . . . . . 8 | |
10 | un0 3576 | . . . . . . . 8 | |
11 | 9, 10 | eqtri 2373 | . . . . . . 7 |
12 | 11 | sseq1i 3296 | . . . . . 6 |
13 | uncom 3409 | . . . . . . 7 | |
14 | 13 | sseq2i 3297 | . . . . . 6 |
15 | 12, 14 | anbi12i 678 | . . . . 5 |
16 | ssunpr 3869 | . . . . 5 | |
17 | uncom 3409 | . . . . . . . . 9 | |
18 | df-pr 3743 | . . . . . . . . 9 | |
19 | 17, 18 | eqtr4i 2376 | . . . . . . . 8 |
20 | 19 | eqeq2i 2363 | . . . . . . 7 |
21 | 20 | orbi2i 505 | . . . . . 6 |
22 | uncom 3409 | . . . . . . . . 9 | |
23 | df-pr 3743 | . . . . . . . . 9 | |
24 | 22, 23 | eqtr4i 2376 | . . . . . . . 8 |
25 | 24 | eqeq2i 2363 | . . . . . . 7 |
26 | 1, 13 | eqtr2i 2374 | . . . . . . . 8 |
27 | 26 | eqeq2i 2363 | . . . . . . 7 |
28 | 25, 27 | orbi12i 507 | . . . . . 6 |
29 | 21, 28 | orbi12i 507 | . . . . 5 |
30 | 15, 16, 29 | 3bitri 262 | . . . 4 |
31 | 8, 30 | orbi12i 507 | . . 3 |
32 | 5, 31 | bitri 240 | . 2 |
33 | 2, 4, 32 | 3bitri 262 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wb 176 wo 357 wa 358 wceq 1642 cun 3208 wss 3258 c0 3551 csn 3738 cpr 3739 ctp 3740 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 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 2479 df-ne 2519 df-ral 2620 df-v 2862 df-nin 3212 df-compl 3213 df-in 3214 df-un 3215 df-dif 3216 df-ss 3260 df-nul 3552 df-sn 3742 df-pr 3743 df-tp 3744 |
This theorem is referenced by: pwtp 3885 |
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