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Mirrors > Home > ILE Home > Th. List > phplem4on | Unicode version |
Description: Equinumerosity of successors of an ordinal and a natural number implies equinumerosity of the originals. (Contributed by Jim Kingdon, 5-Sep-2021.) |
Ref | Expression |
---|---|
phplem4on |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bren 6641 | . . . . 5 | |
2 | 1 | biimpi 119 | . . . 4 |
3 | 2 | adantl 275 | . . 3 |
4 | f1of1 5366 | . . . . . . . 8 | |
5 | 4 | adantl 275 | . . . . . . 7 |
6 | peano2 4509 | . . . . . . . . 9 | |
7 | nnon 4523 | . . . . . . . . 9 | |
8 | 6, 7 | syl 14 | . . . . . . . 8 |
9 | 8 | ad3antlr 484 | . . . . . . 7 |
10 | sssucid 4337 | . . . . . . . 8 | |
11 | 10 | a1i 9 | . . . . . . 7 |
12 | simplll 522 | . . . . . . 7 | |
13 | f1imaen2g 6687 | . . . . . . 7 | |
14 | 5, 9, 11, 12, 13 | syl22anc 1217 | . . . . . 6 |
15 | 14 | ensymd 6677 | . . . . 5 |
16 | eloni 4297 | . . . . . . . . 9 | |
17 | orddif 4462 | . . . . . . . . 9 | |
18 | 16, 17 | syl 14 | . . . . . . . 8 |
19 | 18 | imaeq2d 4881 | . . . . . . 7 |
20 | 19 | ad3antrrr 483 | . . . . . 6 |
21 | f1ofn 5368 | . . . . . . . . . 10 | |
22 | 21 | adantl 275 | . . . . . . . . 9 |
23 | sucidg 4338 | . . . . . . . . . 10 | |
24 | 12, 23 | syl 14 | . . . . . . . . 9 |
25 | fnsnfv 5480 | . . . . . . . . 9 | |
26 | 22, 24, 25 | syl2anc 408 | . . . . . . . 8 |
27 | 26 | difeq2d 3194 | . . . . . . 7 |
28 | imadmrn 4891 | . . . . . . . . . . 11 | |
29 | 28 | eqcomi 2143 | . . . . . . . . . 10 |
30 | f1ofo 5374 | . . . . . . . . . . 11 | |
31 | forn 5348 | . . . . . . . . . . 11 | |
32 | 30, 31 | syl 14 | . . . . . . . . . 10 |
33 | f1odm 5371 | . . . . . . . . . . 11 | |
34 | 33 | imaeq2d 4881 | . . . . . . . . . 10 |
35 | 29, 32, 34 | 3eqtr3a 2196 | . . . . . . . . 9 |
36 | 35 | difeq1d 3193 | . . . . . . . 8 |
37 | 36 | adantl 275 | . . . . . . 7 |
38 | dff1o3 5373 | . . . . . . . . . 10 | |
39 | 38 | simprbi 273 | . . . . . . . . 9 |
40 | imadif 5203 | . . . . . . . . 9 | |
41 | 39, 40 | syl 14 | . . . . . . . 8 |
42 | 41 | adantl 275 | . . . . . . 7 |
43 | 27, 37, 42 | 3eqtr4rd 2183 | . . . . . 6 |
44 | 20, 43 | eqtrd 2172 | . . . . 5 |
45 | 15, 44 | breqtrd 3954 | . . . 4 |
46 | simpllr 523 | . . . . . 6 | |
47 | fnfvelrn 5552 | . . . . . . . 8 | |
48 | 22, 24, 47 | syl2anc 408 | . . . . . . 7 |
49 | 31 | eleq2d 2209 | . . . . . . . . 9 |
50 | 30, 49 | syl 14 | . . . . . . . 8 |
51 | 50 | adantl 275 | . . . . . . 7 |
52 | 48, 51 | mpbid 146 | . . . . . 6 |
53 | phplem3g 6750 | . . . . . 6 | |
54 | 46, 52, 53 | syl2anc 408 | . . . . 5 |
55 | 54 | ensymd 6677 | . . . 4 |
56 | entr 6678 | . . . 4 | |
57 | 45, 55, 56 | syl2anc 408 | . . 3 |
58 | 3, 57 | exlimddv 1870 | . 2 |
59 | 58 | ex 114 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wex 1468 wcel 1480 cdif 3068 wss 3071 csn 3527 class class class wbr 3929 word 4284 con0 4285 csuc 4287 com 4504 ccnv 4538 cdm 4539 crn 4540 cima 4542 wfun 5117 wfn 5118 wf1 5120 wfo 5121 wf1o 5122 cfv 5123 cen 6632 |
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-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-er 6429 df-en 6635 |
This theorem is referenced by: (None) |
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