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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 6691 | . . . . . 6 | |
3 | 1, 2 | syl 14 | . . . . 5 |
4 | 3 | simpld 111 | . . . 4 |
5 | abid2 2278 | . . . . . 6 | |
6 | vex 2715 | . . . . . 6 | |
7 | 5, 6 | eqeltri 2230 | . . . . 5 |
8 | 7 | a1i 9 | . . . 4 |
9 | 4, 8 | opabex3d 6069 | . . 3 |
10 | ccfunen.cc | . . . 4 CCHOICE | |
11 | df-cc 7183 | . . . 4 CCHOICE | |
12 | 10, 11 | sylib 121 | . . 3 |
13 | ccfunen.m | . . . . . 6 | |
14 | elequ2 2133 | . . . . . . . . 9 | |
15 | 14 | exbidv 1805 | . . . . . . . 8 |
16 | 15 | cbvralv 2680 | . . . . . . 7 |
17 | elequ1 2132 | . . . . . . . . 9 | |
18 | 17 | cbvexv 1898 | . . . . . . . 8 |
19 | 18 | ralbii 2463 | . . . . . . 7 |
20 | 16, 19 | bitri 183 | . . . . . 6 |
21 | 13, 20 | sylib 121 | . . . . 5 |
22 | dmopab3 4799 | . . . . 5 | |
23 | 21, 22 | sylib 121 | . . . 4 |
24 | 23, 1 | eqbrtrd 3986 | . . 3 |
25 | dmeq 4786 | . . . . . 6 | |
26 | 25 | breq1d 3975 | . . . . 5 |
27 | sseq2 3152 | . . . . . . 7 | |
28 | 25 | fneq2d 5261 | . . . . . . 7 |
29 | 27, 28 | anbi12d 465 | . . . . . 6 |
30 | 29 | exbidv 1805 | . . . . 5 |
31 | 26, 30 | imbi12d 233 | . . . 4 |
32 | 31 | spcgv 2799 | . . 3 |
33 | 9, 12, 24, 32 | syl3c 63 | . 2 |
34 | simprr 522 | . . . . . 6 | |
35 | 23 | fneq2d 5261 | . . . . . . 7 |
36 | 35 | adantr 274 | . . . . . 6 |
37 | 34, 36 | mpbid 146 | . . . . 5 |
38 | simplrl 525 | . . . . . . . . 9 | |
39 | fnopfv 5597 | . . . . . . . . . 10 | |
40 | 37, 39 | sylan 281 | . . . . . . . . 9 |
41 | 38, 40 | sseldd 3129 | . . . . . . . 8 |
42 | vex 2715 | . . . . . . . . 9 | |
43 | vex 2715 | . . . . . . . . . 10 | |
44 | 43, 42 | fvex 5488 | . . . . . . . . 9 |
45 | eleq1 2220 | . . . . . . . . . 10 | |
46 | elequ2 2133 | . . . . . . . . . 10 | |
47 | 45, 46 | anbi12d 465 | . . . . . . . . 9 |
48 | eleq1 2220 | . . . . . . . . . 10 | |
49 | 48 | anbi2d 460 | . . . . . . . . 9 |
50 | 42, 44, 47, 49 | opelopab 4231 | . . . . . . . 8 |
51 | 41, 50 | sylib 121 | . . . . . . 7 |
52 | 51 | simprd 113 | . . . . . 6 |
53 | 52 | ralrimiva 2530 | . . . . 5 |
54 | 37, 53 | jca 304 | . . . 4 |
55 | 54 | ex 114 | . . 3 |
56 | 55 | eximdv 1860 | . 2 |
57 | 33, 56 | mpd 13 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wal 1333 wceq 1335 wex 1472 wcel 2128 cab 2143 wral 2435 cvv 2712 wss 3102 cop 3563 class class class wbr 3965 copab 4024 com 4549 cdm 4586 wfn 5165 cfv 5170 cen 6683 CCHOICEwacc 7182 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-en 6686 df-cc 7183 |
This theorem is referenced by: cc1 7185 cc2lem 7186 |
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