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Mirrors > Home > ILE Home > Th. List > phplem4 | Unicode version |
Description: Lemma for Pigeonhole Principle. Equinumerosity of successors implies equinumerosity of the original natural numbers. (Contributed by NM, 28-May-1998.) (Revised by Mario Carneiro, 24-Jun-2015.) |
Ref | Expression |
---|---|
phplem2.1 | |
phplem2.2 |
Ref | Expression |
---|---|
phplem4 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bren 6641 | . 2 | |
2 | f1of1 5366 | . . . . . . . . . 10 | |
3 | 2 | adantl 275 | . . . . . . . . 9 |
4 | phplem2.2 | . . . . . . . . . 10 | |
5 | 4 | sucex 4415 | . . . . . . . . 9 |
6 | sssucid 4337 | . . . . . . . . . 10 | |
7 | phplem2.1 | . . . . . . . . . 10 | |
8 | f1imaen2g 6687 | . . . . . . . . . 10 | |
9 | 6, 7, 8 | mpanr12 435 | . . . . . . . . 9 |
10 | 3, 5, 9 | sylancl 409 | . . . . . . . 8 |
11 | 10 | ensymd 6677 | . . . . . . 7 |
12 | nnord 4525 | . . . . . . . . . 10 | |
13 | orddif 4462 | . . . . . . . . . 10 | |
14 | 12, 13 | syl 14 | . . . . . . . . 9 |
15 | 14 | imaeq2d 4881 | . . . . . . . 8 |
16 | f1ofn 5368 | . . . . . . . . . . 11 | |
17 | 7 | sucid 4339 | . . . . . . . . . . 11 |
18 | fnsnfv 5480 | . . . . . . . . . . 11 | |
19 | 16, 17, 18 | sylancl 409 | . . . . . . . . . 10 |
20 | 19 | difeq2d 3194 | . . . . . . . . 9 |
21 | imadmrn 4891 | . . . . . . . . . . . 12 | |
22 | 21 | eqcomi 2143 | . . . . . . . . . . 11 |
23 | f1ofo 5374 | . . . . . . . . . . . 12 | |
24 | forn 5348 | . . . . . . . . . . . 12 | |
25 | 23, 24 | syl 14 | . . . . . . . . . . 11 |
26 | f1odm 5371 | . . . . . . . . . . . 12 | |
27 | 26 | imaeq2d 4881 | . . . . . . . . . . 11 |
28 | 22, 25, 27 | 3eqtr3a 2196 | . . . . . . . . . 10 |
29 | 28 | difeq1d 3193 | . . . . . . . . 9 |
30 | dff1o3 5373 | . . . . . . . . . . 11 | |
31 | 30 | simprbi 273 | . . . . . . . . . 10 |
32 | imadif 5203 | . . . . . . . . . 10 | |
33 | 31, 32 | syl 14 | . . . . . . . . 9 |
34 | 20, 29, 33 | 3eqtr4rd 2183 | . . . . . . . 8 |
35 | 15, 34 | sylan9eq 2192 | . . . . . . 7 |
36 | 11, 35 | breqtrd 3954 | . . . . . 6 |
37 | fnfvelrn 5552 | . . . . . . . . . 10 | |
38 | 16, 17, 37 | sylancl 409 | . . . . . . . . 9 |
39 | 24 | eleq2d 2209 | . . . . . . . . . 10 |
40 | 23, 39 | syl 14 | . . . . . . . . 9 |
41 | 38, 40 | mpbid 146 | . . . . . . . 8 |
42 | vex 2689 | . . . . . . . . . 10 | |
43 | 42, 7 | fvex 5441 | . . . . . . . . 9 |
44 | 4, 43 | phplem3 6748 | . . . . . . . 8 |
45 | 41, 44 | sylan2 284 | . . . . . . 7 |
46 | 45 | ensymd 6677 | . . . . . 6 |
47 | entr 6678 | . . . . . 6 | |
48 | 36, 46, 47 | syl2an 287 | . . . . 5 |
49 | 48 | anandirs 582 | . . . 4 |
50 | 49 | ex 114 | . . 3 |
51 | 50 | exlimdv 1791 | . 2 |
52 | 1, 51 | syl5bi 151 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wex 1468 wcel 1480 cvv 2686 cdif 3068 wss 3071 csn 3527 class class class wbr 3929 word 4284 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: nneneq 6751 php5 6752 |
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