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Mirrors > Home > ILE Home > Th. List > phpelm | Unicode version |
Description: Pigeonhole Principle. A natural number is not equinumerous to an element of itself. (Contributed by Jim Kingdon, 6-Sep-2021.) |
Ref | Expression |
---|---|
phpelm |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 108 | . 2 | |
2 | nnon 4523 | . . . 4 | |
3 | onelss 4309 | . . . 4 | |
4 | 2, 3 | syl 14 | . . 3 |
5 | 4 | imp 123 | . 2 |
6 | simpr 109 | . . 3 | |
7 | elirr 4456 | . . . . 5 | |
8 | 7 | a1i 9 | . . . 4 |
9 | 6, 8 | eldifd 3081 | . . 3 |
10 | eleq1 2202 | . . . 4 | |
11 | 10 | spcegv 2774 | . . 3 |
12 | 6, 9, 11 | sylc 62 | . 2 |
13 | phpm 6759 | . 2 | |
14 | 1, 5, 12, 13 | syl3anc 1216 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wex 1468 wcel 1480 cdif 3068 wss 3071 class class class wbr 3929 con0 4285 com 4504 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 df-dom 6636 |
This theorem is referenced by: (None) |
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