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Mirrors > Home > NFE Home > Th. List > opkabssvvk | Unicode version |
Description: Any Kuratowski ordered pair abstraction is a subset of k . (Contributed by SF, 13-Jan-2015.) |
Ref | Expression |
---|---|
opkabssvvk | k |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2353 | . . . . . . 7 | |
2 | vex 2862 | . . . . . . . 8 | |
3 | vex 2862 | . . . . . . . 8 | |
4 | opkeq12 4061 | . . . . . . . . 9 | |
5 | 4 | eqeq2d 2364 | . . . . . . . 8 |
6 | 2, 3, 5 | spc2ev 2947 | . . . . . . 7 |
7 | 1, 6 | ax-mp 8 | . . . . . 6 |
8 | elvvk 4207 | . . . . . 6 k | |
9 | 7, 8 | mpbir 200 | . . . . 5 k |
10 | eleq1 2413 | . . . . 5 k k | |
11 | 9, 10 | mpbiri 224 | . . . 4 k |
12 | 11 | adantr 451 | . . 3 k |
13 | 12 | exlimivv 1635 | . 2 k |
14 | 13 | abssi 3341 | 1 k |
Colors of variables: wff setvar class |
Syntax hints: wa 358 wex 1541 wceq 1642 wcel 1710 cab 2339 cvv 2859 wss 3257 copk 4057 k cxpk 4174 |
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 ax-nin 4078 ax-sn 4087 |
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-ne 2518 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-ss 3259 df-nul 3551 df-sn 3741 df-pr 3742 df-opk 4058 df-xpk 4185 |
This theorem is referenced by: opkabssvvki 4209 |
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