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Mirrors > Home > MPE Home > Th. List > karden | Structured version Visualization version Unicode version |
Description: If we allow the Axiom of Regularity, we can avoid the Axiom of Choice by defining the cardinal number of a set as the set of all sets equinumerous to it and having the least possible rank. This theorem proves the equinumerosity relationship for this definition (compare carden 9373). The hypotheses correspond to the definition of kard of [Enderton] p. 222 (which we don't define separately since currently we do not use it elsewhere). This theorem along with kardex 8757 justify the definition of kard. The restriction to the least rank prevents the proper class that would result from . (Contributed by NM, 18-Dec-2003.) |
Ref | Expression |
---|---|
karden.1 | |
karden.2 | |
karden.3 | |
karden.4 |
Ref | Expression |
---|---|
karden |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | karden.1 | . . . . . . . 8 | |
2 | 1 | enref 7988 | . . . . . . 7 |
3 | breq1 4656 | . . . . . . . 8 | |
4 | 1, 3 | spcev 3300 | . . . . . . 7 |
5 | 2, 4 | ax-mp 5 | . . . . . 6 |
6 | abn0 3954 | . . . . . 6 | |
7 | 5, 6 | mpbir 221 | . . . . 5 |
8 | scott0 8749 | . . . . . 6 | |
9 | 8 | necon3bii 2846 | . . . . 5 |
10 | 7, 9 | mpbi 220 | . . . 4 |
11 | rabn0 3958 | . . . 4 | |
12 | 10, 11 | mpbi 220 | . . 3 |
13 | vex 3203 | . . . . . . . 8 | |
14 | breq1 4656 | . . . . . . . 8 | |
15 | 13, 14 | elab 3350 | . . . . . . 7 |
16 | breq1 4656 | . . . . . . . 8 | |
17 | 16 | ralab 3367 | . . . . . . 7 |
18 | 15, 17 | anbi12i 733 | . . . . . 6 |
19 | simpl 473 | . . . . . . . . 9 | |
20 | 19 | a1i 11 | . . . . . . . 8 |
21 | karden.3 | . . . . . . . . . . . 12 | |
22 | karden.4 | . . . . . . . . . . . 12 | |
23 | 21, 22 | eqeq12i 2636 | . . . . . . . . . . 11 |
24 | abbi 2737 | . . . . . . . . . . 11 | |
25 | 23, 24 | bitr4i 267 | . . . . . . . . . 10 |
26 | breq1 4656 | . . . . . . . . . . . . 13 | |
27 | fveq2 6191 | . . . . . . . . . . . . . . . 16 | |
28 | 27 | sseq1d 3632 | . . . . . . . . . . . . . . 15 |
29 | 28 | imbi2d 330 | . . . . . . . . . . . . . 14 |
30 | 29 | albidv 1849 | . . . . . . . . . . . . 13 |
31 | 26, 30 | anbi12d 747 | . . . . . . . . . . . 12 |
32 | breq1 4656 | . . . . . . . . . . . . 13 | |
33 | 28 | imbi2d 330 | . . . . . . . . . . . . . 14 |
34 | 33 | albidv 1849 | . . . . . . . . . . . . 13 |
35 | 32, 34 | anbi12d 747 | . . . . . . . . . . . 12 |
36 | 31, 35 | bibi12d 335 | . . . . . . . . . . 11 |
37 | 36 | spv 2260 | . . . . . . . . . 10 |
38 | 25, 37 | sylbi 207 | . . . . . . . . 9 |
39 | simpl 473 | . . . . . . . . 9 | |
40 | 38, 39 | syl6bi 243 | . . . . . . . 8 |
41 | 20, 40 | jcad 555 | . . . . . . 7 |
42 | ensym 8005 | . . . . . . . 8 | |
43 | entr 8008 | . . . . . . . 8 | |
44 | 42, 43 | sylan 488 | . . . . . . 7 |
45 | 41, 44 | syl6 35 | . . . . . 6 |
46 | 18, 45 | syl5bi 232 | . . . . 5 |
47 | 46 | expd 452 | . . . 4 |
48 | 47 | rexlimdv 3030 | . . 3 |
49 | 12, 48 | mpi 20 | . 2 |
50 | enen2 8101 | . . . . 5 | |
51 | enen2 8101 | . . . . . . 7 | |
52 | 51 | imbi1d 331 | . . . . . 6 |
53 | 52 | albidv 1849 | . . . . 5 |
54 | 50, 53 | anbi12d 747 | . . . 4 |
55 | 54 | abbidv 2741 | . . 3 |
56 | 55, 21, 22 | 3eqtr4g 2681 | . 2 |
57 | 49, 56 | impbii 199 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wal 1481 wceq 1483 wex 1704 wcel 1990 cab 2608 wne 2794 wral 2912 wrex 2913 crab 2916 cvv 3200 wss 3574 c0 3915 class class class wbr 4653 cfv 5888 cen 7952 crnk 8626 |
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-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-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-int 4476 df-iun 4522 df-iin 4523 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-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-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-r1 8627 df-rank 8628 |
This theorem is referenced by: (None) |
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