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Mirrors > Home > ILE Home > Th. List > phplem2 | Unicode version |
Description: Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus one of its elements. (Contributed by NM, 11-Jun-1998.) (Revised by Mario Carneiro, 16-Nov-2014.) |
Ref | Expression |
---|---|
phplem2.1 | |
phplem2.2 |
Ref | Expression |
---|---|
phplem2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | phplem2.2 | . . . . . . . 8 | |
2 | phplem2.1 | . . . . . . . 8 | |
3 | 1, 2 | opex 4151 | . . . . . . 7 |
4 | 3 | snex 4109 | . . . . . 6 |
5 | 1, 2 | f1osn 5407 | . . . . . 6 |
6 | f1oen3g 6648 | . . . . . 6 | |
7 | 4, 5, 6 | mp2an 422 | . . . . 5 |
8 | difss 3202 | . . . . . . 7 | |
9 | 2, 8 | ssexi 4066 | . . . . . 6 |
10 | 9 | enref 6659 | . . . . 5 |
11 | 7, 10 | pm3.2i 270 | . . . 4 |
12 | incom 3268 | . . . . . 6 | |
13 | ssrin 3301 | . . . . . . . . 9 | |
14 | 8, 13 | ax-mp 5 | . . . . . . . 8 |
15 | nnord 4525 | . . . . . . . . 9 | |
16 | orddisj 4461 | . . . . . . . . 9 | |
17 | 15, 16 | syl 14 | . . . . . . . 8 |
18 | 14, 17 | sseqtrid 3147 | . . . . . . 7 |
19 | ss0 3403 | . . . . . . 7 | |
20 | 18, 19 | syl 14 | . . . . . 6 |
21 | 12, 20 | syl5eq 2184 | . . . . 5 |
22 | disjdif 3435 | . . . . 5 | |
23 | 21, 22 | jctil 310 | . . . 4 |
24 | unen 6710 | . . . 4 | |
25 | 11, 23, 24 | sylancr 410 | . . 3 |
26 | 25 | adantr 274 | . 2 |
27 | uncom 3220 | . . 3 | |
28 | nndifsnid 6403 | . . 3 | |
29 | 27, 28 | syl5eq 2184 | . 2 |
30 | phplem1 6746 | . 2 | |
31 | 26, 29, 30 | 3brtr3d 3959 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 cvv 2686 cdif 3068 cun 3069 cin 3070 wss 3071 c0 3363 csn 3527 cop 3530 class class class wbr 3929 word 4284 csuc 4287 com 4504 wf1o 5122 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-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-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-en 6635 |
This theorem is referenced by: phplem3 6748 |
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