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Mirrors > Home > ILE Home > Th. List > snexxph | Unicode version |
Description: A case where the antecedent of snexg 4078 is not needed. The class is from dcextest 4465. (Contributed by Mario Carneiro and Jim Kingdon, 4-Jul-2022.) |
Ref | Expression |
---|---|
snexxph |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1on 6288 | . . 3 | |
2 | 1 | elexi 2672 | . 2 |
3 | elsni 3515 | . . . . 5 | |
4 | vprc 4030 | . . . . . . . 8 | |
5 | df-v 2662 | . . . . . . . . . 10 | |
6 | equid 1662 | . . . . . . . . . . . 12 | |
7 | pm5.1im 172 | . . . . . . . . . . . 12 | |
8 | 6, 7 | ax-mp 5 | . . . . . . . . . . 11 |
9 | 8 | abbidv 2235 | . . . . . . . . . 10 |
10 | 5, 9 | syl5req 2163 | . . . . . . . . 9 |
11 | 10 | eleq1d 2186 | . . . . . . . 8 |
12 | 4, 11 | mtbiri 649 | . . . . . . 7 |
13 | 19.8a 1554 | . . . . . . . . 9 | |
14 | 3, 13 | syl 14 | . . . . . . . 8 |
15 | isset 2666 | . . . . . . . 8 | |
16 | 14, 15 | sylibr 133 | . . . . . . 7 |
17 | 12, 16 | nsyl3 600 | . . . . . 6 |
18 | vex 2663 | . . . . . . . . . 10 | |
19 | biidd 171 | . . . . . . . . . 10 | |
20 | 18, 19 | elab 2802 | . . . . . . . . 9 |
21 | 20 | notbii 642 | . . . . . . . 8 |
22 | 21 | biimpri 132 | . . . . . . 7 |
23 | 22 | eq0rdv 3377 | . . . . . 6 |
24 | 17, 23 | syl 14 | . . . . 5 |
25 | 3, 24 | eqtrd 2150 | . . . 4 |
26 | 0lt1o 6305 | . . . 4 | |
27 | 25, 26 | syl6eqel 2208 | . . 3 |
28 | 27 | ssriv 3071 | . 2 |
29 | 2, 28 | ssexi 4036 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wb 104 wceq 1316 wex 1453 wcel 1465 cab 2103 cvv 2660 c0 3333 csn 3497 con0 4255 c1o 6274 |
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-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-uni 3707 df-tr 3997 df-iord 4258 df-on 4260 df-suc 4263 df-1o 6281 |
This theorem is referenced by: (None) |
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