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Mirrors > Home > ILE Home > Th. List > phplem3 | Unicode version |
Description: Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus any element of the successor. For a version without the redundant hypotheses, see phplem3g 6319. (Contributed by NM, 26-May-1998.) |
Ref | Expression |
---|---|
phplem2.1 | |
phplem2.2 |
Ref | Expression |
---|---|
phplem3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elsuci 4140 | . 2 | |
2 | phplem2.1 | . . . 4 | |
3 | phplem2.2 | . . . 4 | |
4 | 2, 3 | phplem2 6316 | . . 3 |
5 | 2 | enref 6245 | . . . 4 |
6 | nnord 4334 | . . . . . 6 | |
7 | orddif 4271 | . . . . . 6 | |
8 | 6, 7 | syl 14 | . . . . 5 |
9 | sneq 3386 | . . . . . . 7 | |
10 | 9 | difeq2d 3062 | . . . . . 6 |
11 | 10 | eqcoms 2043 | . . . . 5 |
12 | 8, 11 | sylan9eq 2092 | . . . 4 |
13 | 5, 12 | syl5breq 3799 | . . 3 |
14 | 4, 13 | jaodan 710 | . 2 |
15 | 1, 14 | sylan2 270 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wo 629 wceq 1243 wcel 1393 cvv 2557 cdif 2914 csn 3375 class class class wbr 3764 word 4099 csuc 4102 com 4313 cen 6219 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-br 3765 df-opab 3819 df-tr 3855 df-id 4030 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-en 6222 |
This theorem is referenced by: phplem4 6318 phplem3g 6319 |
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