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Mirrors > Home > NFE Home > Th. List > ncspw1eu | Unicode version |
Description: Given a cardinal, there is a unique cardinal that contains the unit power class of its members. (Contributed by SF, 2-Mar-2015.) |
Ref | Expression |
---|---|
ncspw1eu | NC NC Nc 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nulnnc 6118 | . . . . . . 7 NC | |
2 | eleq1 2413 | . . . . . . 7 NC NC | |
3 | 1, 2 | mtbiri 294 | . . . . . 6 NC |
4 | 3 | necon2ai 2561 | . . . . 5 NC |
5 | n0 3559 | . . . . 5 | |
6 | 4, 5 | sylib 188 | . . . 4 NC |
7 | vex 2862 | . . . . . . . . . 10 | |
8 | 7 | pw1ex 4303 | . . . . . . . . 9 1 |
9 | 8 | ncelncsi 6121 | . . . . . . . 8 Nc 1 NC |
10 | eqid 2353 | . . . . . . . 8 Nc 1 Nc 1 | |
11 | eqeq1 2359 | . . . . . . . . 9 Nc 1 Nc 1 Nc 1 Nc 1 | |
12 | 11 | rspcev 2955 | . . . . . . . 8 Nc 1 NC Nc 1 Nc 1 NC Nc 1 |
13 | 9, 10, 12 | mp2an 653 | . . . . . . 7 NC Nc 1 |
14 | 13 | jctr 526 | . . . . . 6 NC Nc 1 |
15 | 14 | a1i 10 | . . . . 5 NC NC Nc 1 |
16 | 15 | eximdv 1622 | . . . 4 NC NC Nc 1 |
17 | 6, 16 | mpd 14 | . . 3 NC NC Nc 1 |
18 | rexcom 2772 | . . . 4 NC Nc 1 NC Nc 1 | |
19 | df-rex 2620 | . . . 4 NC Nc 1 NC Nc 1 | |
20 | 18, 19 | bitri 240 | . . 3 NC Nc 1 NC Nc 1 |
21 | 17, 20 | sylibr 203 | . 2 NC NC Nc 1 |
22 | reeanv 2778 | . . . 4 Nc 1 Nc 1 Nc 1 Nc 1 | |
23 | ncseqnc 6128 | . . . . . . . . . . . 12 NC Nc | |
24 | 23 | biimpar 471 | . . . . . . . . . . 11 NC Nc |
25 | 24 | adantrr 697 | . . . . . . . . . 10 NC Nc |
26 | ncseqnc 6128 | . . . . . . . . . . . 12 NC Nc | |
27 | 26 | biimpar 471 | . . . . . . . . . . 11 NC Nc |
28 | 27 | adantrl 696 | . . . . . . . . . 10 NC Nc |
29 | 25, 28 | eqtr3d 2387 | . . . . . . . . 9 NC Nc Nc |
30 | 7 | ncpw1 6152 | . . . . . . . . 9 Nc Nc Nc 1 Nc 1 |
31 | 29, 30 | sylib 188 | . . . . . . . 8 NC Nc 1 Nc 1 |
32 | 31 | 3adant2 974 | . . . . . . 7 NC NC NC Nc 1 Nc 1 |
33 | eqeq2 2362 | . . . . . . . . 9 Nc 1 Nc 1 Nc 1 Nc 1 | |
34 | 33 | anbi1d 685 | . . . . . . . 8 Nc 1 Nc 1 Nc 1 Nc 1 Nc 1 Nc 1 |
35 | eqtr3 2372 | . . . . . . . 8 Nc 1 Nc 1 | |
36 | 34, 35 | syl6bi 219 | . . . . . . 7 Nc 1 Nc 1 Nc 1 Nc 1 |
37 | 32, 36 | syl 15 | . . . . . 6 NC NC NC Nc 1 Nc 1 |
38 | 37 | 3expa 1151 | . . . . 5 NC NC NC Nc 1 Nc 1 |
39 | 38 | rexlimdvva 2745 | . . . 4 NC NC NC Nc 1 Nc 1 |
40 | 22, 39 | syl5bir 209 | . . 3 NC NC NC Nc 1 Nc 1 |
41 | 40 | ralrimivva 2706 | . 2 NC NC NC Nc 1 Nc 1 |
42 | eqeq1 2359 | . . . . 5 Nc 1 Nc 1 | |
43 | 42 | rexbidv 2635 | . . . 4 Nc 1 Nc 1 |
44 | pw1eq 4143 | . . . . . . 7 1 1 | |
45 | 44 | nceqd 6110 | . . . . . 6 Nc 1 Nc 1 |
46 | 45 | eqeq2d 2364 | . . . . 5 Nc 1 Nc 1 |
47 | 46 | cbvrexv 2836 | . . . 4 Nc 1 Nc 1 |
48 | 43, 47 | syl6bb 252 | . . 3 Nc 1 Nc 1 |
49 | 48 | reu4 3030 | . 2 NC Nc 1 NC Nc 1 NC NC Nc 1 Nc 1 |
50 | 21, 41, 49 | sylanbrc 645 | 1 NC NC Nc 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 358 w3a 934 wex 1541 wceq 1642 wcel 1710 wne 2516 wral 2614 wrex 2615 wreu 2616 c0 3550 1 cpw1 4135 NC cncs 6088 Nc cnc 6091 |
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-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-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 |
This theorem is referenced by: tccl 6160 eqtc 6161 |
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