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Mirrors > Home > MPE Home > Th. List > card2on | Structured version Visualization version Unicode version |
Description: Proof that the alternate definition cardval2 8817 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 5748 | . . . . . . . . . . . . 13 | |
2 | vex 3203 | . . . . . . . . . . . . . 14 | |
3 | onelss 5766 | . . . . . . . . . . . . . . 15 | |
4 | 3 | imp 445 | . . . . . . . . . . . . . 14 |
5 | ssdomg 8001 | . . . . . . . . . . . . . 14 | |
6 | 2, 4, 5 | mpsyl 68 | . . . . . . . . . . . . 13 |
7 | 1, 6 | jca 554 | . . . . . . . . . . . 12 |
8 | domsdomtr 8095 | . . . . . . . . . . . . . 14 | |
9 | 8 | anim2i 593 | . . . . . . . . . . . . 13 |
10 | 9 | anassrs 680 | . . . . . . . . . . . 12 |
11 | 7, 10 | sylan 488 | . . . . . . . . . . 11 |
12 | 11 | exp31 630 | . . . . . . . . . 10 |
13 | 12 | com12 32 | . . . . . . . . 9 |
14 | 13 | impd 447 | . . . . . . . 8 |
15 | breq1 4656 | . . . . . . . . 9 | |
16 | 15 | elrab 3363 | . . . . . . . 8 |
17 | breq1 4656 | . . . . . . . . 9 | |
18 | 17 | elrab 3363 | . . . . . . . 8 |
19 | 14, 16, 18 | 3imtr4g 285 | . . . . . . 7 |
20 | 19 | imp 445 | . . . . . 6 |
21 | 20 | gen2 1723 | . . . . 5 |
22 | dftr2 4754 | . . . . 5 | |
23 | 21, 22 | mpbir 221 | . . . 4 |
24 | ssrab2 3687 | . . . 4 | |
25 | ordon 6982 | . . . 4 | |
26 | trssord 5740 | . . . 4 | |
27 | 23, 24, 25, 26 | mp3an 1424 | . . 3 |
28 | hartogs 8449 | . . . 4 | |
29 | sdomdom 7983 | . . . . . . 7 | |
30 | 29 | a1i 11 | . . . . . 6 |
31 | 30 | ss2rabi 3684 | . . . . 5 |
32 | ssexg 4804 | . . . . 5 | |
33 | 31, 32 | mpan 706 | . . . 4 |
34 | elong 5731 | . . . 4 | |
35 | 28, 33, 34 | 3syl 18 | . . 3 |
36 | 27, 35 | mpbiri 248 | . 2 |
37 | 0elon 5778 | . . . 4 | |
38 | eleq1 2689 | . . . 4 | |
39 | 37, 38 | mpbiri 248 | . . 3 |
40 | df-ne 2795 | . . . . 5 | |
41 | rabn0 3958 | . . . . 5 | |
42 | 40, 41 | bitr3i 266 | . . . 4 |
43 | relsdom 7962 | . . . . . 6 | |
44 | 43 | brrelex2i 5159 | . . . . 5 |
45 | 44 | rexlimivw 3029 | . . . 4 |
46 | 42, 45 | sylbi 207 | . . 3 |
47 | 39, 46 | nsyl4 156 | . 2 |
48 | 36, 47 | pm2.61i 176 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wa 384 wal 1481 wceq 1483 wcel 1990 wne 2794 wrex 2913 crab 2916 cvv 3200 wss 3574 c0 3915 class class class wbr 4653 wtr 4752 word 5722 con0 5723 cdom 7953 csdm 7954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-wrecs 7407 df-recs 7468 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-oi 8415 |
This theorem is referenced by: (None) |
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