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Mirrors > Home > MPE Home > Th. List > card2on | Unicode version |
Description: Proof that the alternate definition cardval2 8393 is always a set, and indeed is an ordinal. (Contributed by Mario Carneiro, 14-Jan-2013.) |
Ref | Expression |
---|---|
card2on |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onelon 4908 | . . . . . . . . . . . . 13 | |
2 | vex 3112 | . . . . . . . . . . . . . 14 | |
3 | onelss 4925 | . . . . . . . . . . . . . . 15 | |
4 | 3 | imp 429 | . . . . . . . . . . . . . 14 |
5 | ssdomg 7581 | . . . . . . . . . . . . . 14 | |
6 | 2, 4, 5 | mpsyl 63 | . . . . . . . . . . . . 13 |
7 | 1, 6 | jca 532 | . . . . . . . . . . . 12 |
8 | domsdomtr 7672 | . . . . . . . . . . . . . 14 | |
9 | 8 | anim2i 569 | . . . . . . . . . . . . 13 |
10 | 9 | anassrs 648 | . . . . . . . . . . . 12 |
11 | 7, 10 | sylan 471 | . . . . . . . . . . 11 |
12 | 11 | exp31 604 | . . . . . . . . . 10 |
13 | 12 | com12 31 | . . . . . . . . 9 |
14 | 13 | impd 431 | . . . . . . . 8 |
15 | breq1 4455 | . . . . . . . . 9 | |
16 | 15 | elrab 3257 | . . . . . . . 8 |
17 | breq1 4455 | . . . . . . . . 9 | |
18 | 17 | elrab 3257 | . . . . . . . 8 |
19 | 14, 16, 18 | 3imtr4g 270 | . . . . . . 7 |
20 | 19 | imp 429 | . . . . . 6 |
21 | 20 | gen2 1619 | . . . . 5 |
22 | dftr2 4547 | . . . . 5 | |
23 | 21, 22 | mpbir 209 | . . . 4 |
24 | ssrab2 3584 | . . . 4 | |
25 | ordon 6618 | . . . 4 | |
26 | trssord 4900 | . . . 4 | |
27 | 23, 24, 25, 26 | mp3an 1324 | . . 3 |
28 | hartogs 7990 | . . . 4 | |
29 | sdomdom 7563 | . . . . . . 7 | |
30 | 29 | a1i 11 | . . . . . 6 |
31 | 30 | ss2rabi 3581 | . . . . 5 |
32 | ssexg 4598 | . . . . 5 | |
33 | 31, 32 | mpan 670 | . . . 4 |
34 | elong 4891 | . . . 4 | |
35 | 28, 33, 34 | 3syl 20 | . . 3 |
36 | 27, 35 | mpbiri 233 | . 2 |
37 | 0elon 4936 | . . . 4 | |
38 | eleq1 2529 | . . . 4 | |
39 | 37, 38 | mpbiri 233 | . . 3 |
40 | df-ne 2654 | . . . . 5 | |
41 | rabn0 3805 | . . . . 5 | |
42 | 40, 41 | bitr3i 251 | . . . 4 |
43 | relsdom 7543 | . . . . . 6 | |
44 | 43 | brrelex2i 5046 | . . . . 5 |
45 | 44 | rexlimivw 2946 | . . . 4 |
46 | 42, 45 | sylbi 195 | . . 3 |
47 | 39, 46 | nsyl4 142 | . 2 |
48 | 36, 47 | pm2.61i 164 | 1 |
Colors of variables: wff setvar class |
Syntax hints: -. wn 3 -> wi 4
<-> wb 184 /\ wa 369 A. wal 1393
= wceq 1395 e. wcel 1818 =/= wne 2652
E. wrex 2808 { crab 2811 cvv 3109
C_ wss 3475 c0 3784 class class class wbr 4452
Tr wtr 4545 Ord word 4882 con0 4883 cdom 7534 csdm 7535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1618 ax-4 1631 ax-5 1704 ax-6 1747 ax-7 1790 ax-8 1820 ax-9 1822 ax-10 1837 ax-11 1842 ax-12 1854 ax-13 1999 ax-ext 2435 ax-rep 4563 ax-sep 4573 ax-nul 4581 ax-pow 4630 ax-pr 4691 ax-un 6592 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 974 df-3an 975 df-tru 1398 df-ex 1613 df-nf 1617 df-sb 1740 df-eu 2286 df-mo 2287 df-clab 2443 df-cleq 2449 df-clel 2452 df-nfc 2607 df-ne 2654 df-ral 2812 df-rex 2813 df-reu 2814 df-rmo 2815 df-rab 2816 df-v 3111 df-sbc 3328 df-csb 3435 df-dif 3478 df-un 3480 df-in 3482 df-ss 3489 df-pss 3491 df-nul 3785 df-if 3942 df-pw 4014 df-sn 4030 df-pr 4032 df-tp 4034 df-op 4036 df-uni 4250 df-iun 4332 df-br 4453 df-opab 4511 df-mpt 4512 df-tr 4546 df-eprel 4796 df-id 4800 df-po 4805 df-so 4806 df-fr 4843 df-se 4844 df-we 4845 df-ord 4886 df-on 4887 df-lim 4888 df-suc 4889 df-xp 5010 df-rel 5011 df-cnv 5012 df-co 5013 df-dm 5014 df-rn 5015 df-res 5016 df-ima 5017 df-iota 5556 df-fun 5595 df-fn 5596 df-f 5597 df-f1 5598 df-fo 5599 df-f1o 5600 df-fv 5601 df-isom 5602 df-riota 6257 df-recs 7061 df-er 7330 df-en 7537 df-dom 7538 df-sdom 7539 df-oi 7956 |
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