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Mirrors > Home > NFE Home > Th. List > el1c | Unicode version |
Description: Membership in cardinal one. (Contributed by SF, 12-Jan-2015.) |
Ref | Expression |
---|---|
el1c | 1c |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2867 | . 2 1c | |
2 | snex 4111 | . . . 4 | |
3 | eleq1 2413 | . . . 4 | |
4 | 2, 3 | mpbiri 224 | . . 3 |
5 | 4 | exlimiv 1634 | . 2 |
6 | eqeq1 2359 | . . . 4 | |
7 | 6 | exbidv 1626 | . . 3 |
8 | df-1c 4136 | . . 3 1c | |
9 | 7, 8 | elab2g 2987 | . 2 1c |
10 | 1, 5, 9 | pm5.21nii 342 | 1 1c |
Colors of variables: wff setvar class |
Syntax hints: wb 176 wex 1541 wceq 1642 wcel 1710 cvv 2859 csn 3737 1cc1c 4134 |
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 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-1c 4136 |
This theorem is referenced by: snel1c 4140 elpw1 4144 elpw11c 4147 0nel1c 4159 eqpw1 4162 df1c2 4168 pw111 4170 eluni1g 4172 opkelimagekg 4271 sikexlem 4295 dfimak2 4298 dfpw2 4327 eqpw1uni 4330 pw1eqadj 4332 dfeu2 4333 dfnnc2 4395 0nelsuc 4400 elsuc 4413 nnsucelrlem1 4424 nnsucelr 4428 ssfin 4470 nnadjoinlem1 4519 sfintfinlem1 4531 spfinex 4537 elimapw11c 4948 pw1fnf1o 5855 1cnc 6139 el2c 6191 |
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