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Mirrors > Home > NFE Home > Th. List > el2c | Unicode version |
Description: Membership in cardinal two. (Contributed by SF, 3-Mar-2015.) |
Ref | Expression |
---|---|
el2c | 2c |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elsuc 4413 | . . 3 1c 1c 1c ∼ | |
2 | df-rex 2620 | . . 3 1c ∼ 1c ∼ | |
3 | el1c 4139 | . . . . . . 7 1c | |
4 | 3 | anbi1i 676 | . . . . . 6 1c ∼ ∼ |
5 | 19.41v 1901 | . . . . . 6 ∼ ∼ | |
6 | 4, 5 | bitr4i 243 | . . . . 5 1c ∼ ∼ |
7 | 6 | exbii 1582 | . . . 4 1c ∼ ∼ |
8 | excom 1741 | . . . 4 ∼ ∼ | |
9 | 7, 8 | bitri 240 | . . 3 1c ∼ ∼ |
10 | 1, 2, 9 | 3bitri 262 | . 2 1c 1c ∼ |
11 | 1p1e2c 6155 | . . 3 1c 1c 2c | |
12 | 11 | eleq2i 2417 | . 2 1c 1c 2c |
13 | snex 4111 | . . . . 5 | |
14 | compleq 3243 | . . . . . 6 ∼ ∼ | |
15 | uneq1 3411 | . . . . . . . 8 | |
16 | df-pr 3742 | . . . . . . . 8 | |
17 | 15, 16 | syl6eqr 2403 | . . . . . . 7 |
18 | 17 | eqeq2d 2364 | . . . . . 6 |
19 | 14, 18 | rexeqbidv 2820 | . . . . 5 ∼ ∼ |
20 | 13, 19 | ceqsexv 2894 | . . . 4 ∼ ∼ |
21 | df-rex 2620 | . . . 4 ∼ ∼ | |
22 | elsn 3748 | . . . . . . . . 9 | |
23 | equcom 1680 | . . . . . . . . 9 | |
24 | 22, 23 | bitri 240 | . . . . . . . 8 |
25 | 24 | notbii 287 | . . . . . . 7 |
26 | vex 2862 | . . . . . . . 8 | |
27 | 26 | elcompl 3225 | . . . . . . 7 ∼ |
28 | df-ne 2518 | . . . . . . 7 | |
29 | 25, 27, 28 | 3bitr4i 268 | . . . . . 6 ∼ |
30 | 29 | anbi1i 676 | . . . . 5 ∼ |
31 | 30 | exbii 1582 | . . . 4 ∼ |
32 | 20, 21, 31 | 3bitri 262 | . . 3 ∼ |
33 | 32 | exbii 1582 | . 2 ∼ |
34 | 10, 12, 33 | 3bitr3i 266 | 1 2c |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wb 176 wa 358 wex 1541 wceq 1642 wcel 1710 wne 2516 wrex 2615 ∼ ccompl 3205 cun 3207 csn 3737 cpr 3738 1cc1c 4134 cplc 4375 2cc2c 6094 |
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-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-1st 4723 df-swap 4724 df-sset 4725 df-co 4726 df-ima 4727 df-si 4728 df-id 4767 df-xp 4784 df-cnv 4785 df-rn 4786 df-dm 4787 df-res 4788 df-fun 4789 df-fn 4790 df-f 4791 df-f1 4792 df-fo 4793 df-f1o 4794 df-fv 4795 df-2nd 4797 df-txp 5736 df-ins2 5750 df-ins3 5752 df-image 5754 df-ins4 5756 df-si3 5758 df-funs 5760 df-fns 5762 df-trans 5899 df-sym 5908 df-er 5909 df-ec 5947 df-qs 5951 df-en 6029 df-ncs 6098 df-nc 6101 df-2c 6104 |
This theorem is referenced by: ce2 6192 |
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