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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 6634 | . . . . 5 | |
2 | 1 | biimpi 119 | . . . 4 |
3 | 2 | adantl 275 | . . 3 |
4 | f1of1 5359 | . . . . . . . 8 | |
5 | 4 | adantl 275 | . . . . . . 7 |
6 | peano2 4504 | . . . . . . . . 9 | |
7 | nnon 4518 | . . . . . . . . 9 | |
8 | 6, 7 | syl 14 | . . . . . . . 8 |
9 | 8 | ad3antlr 484 | . . . . . . 7 |
10 | sssucid 4332 | . . . . . . . 8 | |
11 | 10 | a1i 9 | . . . . . . 7 |
12 | simplll 522 | . . . . . . 7 | |
13 | f1imaen2g 6680 | . . . . . . 7 | |
14 | 5, 9, 11, 12, 13 | syl22anc 1217 | . . . . . 6 |
15 | 14 | ensymd 6670 | . . . . 5 |
16 | eloni 4292 | . . . . . . . . 9 | |
17 | orddif 4457 | . . . . . . . . 9 | |
18 | 16, 17 | syl 14 | . . . . . . . 8 |
19 | 18 | imaeq2d 4876 | . . . . . . 7 |
20 | 19 | ad3antrrr 483 | . . . . . 6 |
21 | f1ofn 5361 | . . . . . . . . . 10 | |
22 | 21 | adantl 275 | . . . . . . . . 9 |
23 | sucidg 4333 | . . . . . . . . . 10 | |
24 | 12, 23 | syl 14 | . . . . . . . . 9 |
25 | fnsnfv 5473 | . . . . . . . . 9 | |
26 | 22, 24, 25 | syl2anc 408 | . . . . . . . 8 |
27 | 26 | difeq2d 3189 | . . . . . . 7 |
28 | imadmrn 4886 | . . . . . . . . . . 11 | |
29 | 28 | eqcomi 2141 | . . . . . . . . . 10 |
30 | f1ofo 5367 | . . . . . . . . . . 11 | |
31 | forn 5343 | . . . . . . . . . . 11 | |
32 | 30, 31 | syl 14 | . . . . . . . . . 10 |
33 | f1odm 5364 | . . . . . . . . . . 11 | |
34 | 33 | imaeq2d 4876 | . . . . . . . . . 10 |
35 | 29, 32, 34 | 3eqtr3a 2194 | . . . . . . . . 9 |
36 | 35 | difeq1d 3188 | . . . . . . . 8 |
37 | 36 | adantl 275 | . . . . . . 7 |
38 | dff1o3 5366 | . . . . . . . . . 10 | |
39 | 38 | simprbi 273 | . . . . . . . . 9 |
40 | imadif 5198 | . . . . . . . . 9 | |
41 | 39, 40 | syl 14 | . . . . . . . 8 |
42 | 41 | adantl 275 | . . . . . . 7 |
43 | 27, 37, 42 | 3eqtr4rd 2181 | . . . . . 6 |
44 | 20, 43 | eqtrd 2170 | . . . . 5 |
45 | 15, 44 | breqtrd 3949 | . . . 4 |
46 | simpllr 523 | . . . . . 6 | |
47 | fnfvelrn 5545 | . . . . . . . 8 | |
48 | 22, 24, 47 | syl2anc 408 | . . . . . . 7 |
49 | 31 | eleq2d 2207 | . . . . . . . . 9 |
50 | 30, 49 | syl 14 | . . . . . . . 8 |
51 | 50 | adantl 275 | . . . . . . 7 |
52 | 48, 51 | mpbid 146 | . . . . . 6 |
53 | phplem3g 6743 | . . . . . 6 | |
54 | 46, 52, 53 | syl2anc 408 | . . . . 5 |
55 | 54 | ensymd 6670 | . . . 4 |
56 | entr 6671 | . . . 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 3063 wss 3066 csn 3522 class class class wbr 3924 word 4279 con0 4280 csuc 4282 com 4499 ccnv 4533 cdm 4534 crn 4535 cima 4537 wfun 5112 wfn 5113 wf1 5115 wfo 5116 wf1o 5117 cfv 5118 cen 6625 |
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 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-er 6422 df-en 6628 |
This theorem is referenced by: (None) |
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