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Mirrors > Home > ILE Home > Th. List > ccfunen | Unicode version |
Description: Existence of a choice function for a countably infinite set. (Contributed by Jim Kingdon, 28-Nov-2023.) |
Ref | Expression |
---|---|
ccfunen.cc | CCHOICE |
ccfunen.a | |
ccfunen.m |
Ref | Expression |
---|---|
ccfunen |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ccfunen.a | . . . . . 6 | |
2 | encv 6724 | . . . . . 6 | |
3 | 1, 2 | syl 14 | . . . . 5 |
4 | 3 | simpld 111 | . . . 4 |
5 | abid2 2291 | . . . . . 6 | |
6 | vex 2733 | . . . . . 6 | |
7 | 5, 6 | eqeltri 2243 | . . . . 5 |
8 | 7 | a1i 9 | . . . 4 |
9 | 4, 8 | opabex3d 6100 | . . 3 |
10 | ccfunen.cc | . . . 4 CCHOICE | |
11 | df-cc 7225 | . . . 4 CCHOICE | |
12 | 10, 11 | sylib 121 | . . 3 |
13 | ccfunen.m | . . . . . 6 | |
14 | elequ2 2146 | . . . . . . . . 9 | |
15 | 14 | exbidv 1818 | . . . . . . . 8 |
16 | 15 | cbvralv 2696 | . . . . . . 7 |
17 | elequ1 2145 | . . . . . . . . 9 | |
18 | 17 | cbvexv 1911 | . . . . . . . 8 |
19 | 18 | ralbii 2476 | . . . . . . 7 |
20 | 16, 19 | bitri 183 | . . . . . 6 |
21 | 13, 20 | sylib 121 | . . . . 5 |
22 | dmopab3 4824 | . . . . 5 | |
23 | 21, 22 | sylib 121 | . . . 4 |
24 | 23, 1 | eqbrtrd 4011 | . . 3 |
25 | dmeq 4811 | . . . . . 6 | |
26 | 25 | breq1d 3999 | . . . . 5 |
27 | sseq2 3171 | . . . . . . 7 | |
28 | 25 | fneq2d 5289 | . . . . . . 7 |
29 | 27, 28 | anbi12d 470 | . . . . . 6 |
30 | 29 | exbidv 1818 | . . . . 5 |
31 | 26, 30 | imbi12d 233 | . . . 4 |
32 | 31 | spcgv 2817 | . . 3 |
33 | 9, 12, 24, 32 | syl3c 63 | . 2 |
34 | simprr 527 | . . . . . 6 | |
35 | 23 | fneq2d 5289 | . . . . . . 7 |
36 | 35 | adantr 274 | . . . . . 6 |
37 | 34, 36 | mpbid 146 | . . . . 5 |
38 | simplrl 530 | . . . . . . . . 9 | |
39 | fnopfv 5626 | . . . . . . . . . 10 | |
40 | 37, 39 | sylan 281 | . . . . . . . . 9 |
41 | 38, 40 | sseldd 3148 | . . . . . . . 8 |
42 | vex 2733 | . . . . . . . . 9 | |
43 | vex 2733 | . . . . . . . . . 10 | |
44 | 43, 42 | fvex 5516 | . . . . . . . . 9 |
45 | eleq1 2233 | . . . . . . . . . 10 | |
46 | elequ2 2146 | . . . . . . . . . 10 | |
47 | 45, 46 | anbi12d 470 | . . . . . . . . 9 |
48 | eleq1 2233 | . . . . . . . . . 10 | |
49 | 48 | anbi2d 461 | . . . . . . . . 9 |
50 | 42, 44, 47, 49 | opelopab 4256 | . . . . . . . 8 |
51 | 41, 50 | sylib 121 | . . . . . . 7 |
52 | 51 | simprd 113 | . . . . . 6 |
53 | 52 | ralrimiva 2543 | . . . . 5 |
54 | 37, 53 | jca 304 | . . . 4 |
55 | 54 | ex 114 | . . 3 |
56 | 55 | eximdv 1873 | . 2 |
57 | 33, 56 | mpd 13 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wal 1346 wceq 1348 wex 1485 wcel 2141 cab 2156 wral 2448 cvv 2730 wss 3121 cop 3586 class class class wbr 3989 copab 4049 com 4574 cdm 4611 wfn 5193 cfv 5198 cen 6716 CCHOICEwacc 7224 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-en 6719 df-cc 7225 |
This theorem is referenced by: cc1 7227 cc2lem 7228 |
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