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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 4189 | . . . . . . 7 |
4 | 3 | snex 4146 | . . . . . 6 |
5 | 1, 2 | f1osn 5454 | . . . . . 6 |
6 | f1oen3g 6699 | . . . . . 6 | |
7 | 4, 5, 6 | mp2an 423 | . . . . 5 |
8 | difss 3233 | . . . . . . 7 | |
9 | 2, 8 | ssexi 4102 | . . . . . 6 |
10 | 9 | enref 6710 | . . . . 5 |
11 | 7, 10 | pm3.2i 270 | . . . 4 |
12 | incom 3299 | . . . . . 6 | |
13 | ssrin 3332 | . . . . . . . . 9 | |
14 | 8, 13 | ax-mp 5 | . . . . . . . 8 |
15 | nnord 4571 | . . . . . . . . 9 | |
16 | orddisj 4505 | . . . . . . . . 9 | |
17 | 15, 16 | syl 14 | . . . . . . . 8 |
18 | 14, 17 | sseqtrid 3178 | . . . . . . 7 |
19 | ss0 3434 | . . . . . . 7 | |
20 | 18, 19 | syl 14 | . . . . . 6 |
21 | 12, 20 | syl5eq 2202 | . . . . 5 |
22 | disjdif 3466 | . . . . 5 | |
23 | 21, 22 | jctil 310 | . . . 4 |
24 | unen 6761 | . . . 4 | |
25 | 11, 23, 24 | sylancr 411 | . . 3 |
26 | 25 | adantr 274 | . 2 |
27 | uncom 3251 | . . 3 | |
28 | nndifsnid 6454 | . . 3 | |
29 | 27, 28 | syl5eq 2202 | . 2 |
30 | phplem1 6797 | . 2 | |
31 | 26, 29, 30 | 3brtr3d 3995 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1335 wcel 2128 cvv 2712 cdif 3099 cun 3100 cin 3101 wss 3102 c0 3394 csn 3560 cop 3563 class class class wbr 3965 word 4322 csuc 4325 com 4549 wf1o 5169 cen 6683 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-br 3966 df-opab 4026 df-tr 4063 df-id 4253 df-iord 4326 df-on 4328 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-en 6686 |
This theorem is referenced by: phplem3 6799 |
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