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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 6737 | . . . . 5 | |
2 | 1 | biimpi 120 | . . . 4 |
3 | 2 | adantl 277 | . . 3 |
4 | f1of1 5452 | . . . . . . . 8 | |
5 | 4 | adantl 277 | . . . . . . 7 |
6 | peano2 4588 | . . . . . . . . 9 | |
7 | nnon 4603 | . . . . . . . . 9 | |
8 | 6, 7 | syl 14 | . . . . . . . 8 |
9 | 8 | ad3antlr 493 | . . . . . . 7 |
10 | sssucid 4409 | . . . . . . . 8 | |
11 | 10 | a1i 9 | . . . . . . 7 |
12 | simplll 533 | . . . . . . 7 | |
13 | f1imaen2g 6783 | . . . . . . 7 | |
14 | 5, 9, 11, 12, 13 | syl22anc 1239 | . . . . . 6 |
15 | 14 | ensymd 6773 | . . . . 5 |
16 | eloni 4369 | . . . . . . . . 9 | |
17 | orddif 4540 | . . . . . . . . 9 | |
18 | 16, 17 | syl 14 | . . . . . . . 8 |
19 | 18 | imaeq2d 4963 | . . . . . . 7 |
20 | 19 | ad3antrrr 492 | . . . . . 6 |
21 | f1ofn 5454 | . . . . . . . . . 10 | |
22 | 21 | adantl 277 | . . . . . . . . 9 |
23 | sucidg 4410 | . . . . . . . . . 10 | |
24 | 12, 23 | syl 14 | . . . . . . . . 9 |
25 | fnsnfv 5567 | . . . . . . . . 9 | |
26 | 22, 24, 25 | syl2anc 411 | . . . . . . . 8 |
27 | 26 | difeq2d 3251 | . . . . . . 7 |
28 | imadmrn 4973 | . . . . . . . . . . 11 | |
29 | 28 | eqcomi 2179 | . . . . . . . . . 10 |
30 | f1ofo 5460 | . . . . . . . . . . 11 | |
31 | forn 5433 | . . . . . . . . . . 11 | |
32 | 30, 31 | syl 14 | . . . . . . . . . 10 |
33 | f1odm 5457 | . . . . . . . . . . 11 | |
34 | 33 | imaeq2d 4963 | . . . . . . . . . 10 |
35 | 29, 32, 34 | 3eqtr3a 2232 | . . . . . . . . 9 |
36 | 35 | difeq1d 3250 | . . . . . . . 8 |
37 | 36 | adantl 277 | . . . . . . 7 |
38 | dff1o3 5459 | . . . . . . . . . 10 | |
39 | 38 | simprbi 275 | . . . . . . . . 9 |
40 | imadif 5288 | . . . . . . . . 9 | |
41 | 39, 40 | syl 14 | . . . . . . . 8 |
42 | 41 | adantl 277 | . . . . . . 7 |
43 | 27, 37, 42 | 3eqtr4rd 2219 | . . . . . 6 |
44 | 20, 43 | eqtrd 2208 | . . . . 5 |
45 | 15, 44 | breqtrd 4024 | . . . 4 |
46 | simpllr 534 | . . . . . 6 | |
47 | fnfvelrn 5640 | . . . . . . . 8 | |
48 | 22, 24, 47 | syl2anc 411 | . . . . . . 7 |
49 | 31 | eleq2d 2245 | . . . . . . . . 9 |
50 | 30, 49 | syl 14 | . . . . . . . 8 |
51 | 50 | adantl 277 | . . . . . . 7 |
52 | 48, 51 | mpbid 147 | . . . . . 6 |
53 | phplem3g 6846 | . . . . . 6 | |
54 | 46, 52, 53 | syl2anc 411 | . . . . 5 |
55 | 54 | ensymd 6773 | . . . 4 |
56 | entr 6774 | . . . 4 | |
57 | 45, 55, 56 | syl2anc 411 | . . 3 |
58 | 3, 57 | exlimddv 1896 | . 2 |
59 | 58 | ex 115 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 104 wb 105 wceq 1353 wex 1490 wcel 2146 cdif 3124 wss 3127 csn 3589 class class class wbr 3998 word 4356 con0 4357 csuc 4359 com 4583 ccnv 4619 cdm 4620 crn 4621 cima 4623 wfun 5202 wfn 5203 wf1 5205 wfo 5206 wf1o 5207 cfv 5208 cen 6728 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-nul 4124 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-iinf 4581 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-ral 2458 df-rex 2459 df-rab 2462 df-v 2737 df-sbc 2961 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-nul 3421 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-br 3999 df-opab 4060 df-tr 4097 df-id 4287 df-iord 4360 df-on 4362 df-suc 4365 df-iom 4584 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-er 6525 df-en 6731 |
This theorem is referenced by: (None) |
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