Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 4214 | . . . . . . 7 |
4 | 3 | snex 4171 | . . . . . 6 |
5 | 1, 2 | f1osn 5482 | . . . . . 6 |
6 | f1oen3g 6732 | . . . . . 6 | |
7 | 4, 5, 6 | mp2an 424 | . . . . 5 |
8 | difss 3253 | . . . . . . 7 | |
9 | 2, 8 | ssexi 4127 | . . . . . 6 |
10 | 9 | enref 6743 | . . . . 5 |
11 | 7, 10 | pm3.2i 270 | . . . 4 |
12 | incom 3319 | . . . . . 6 | |
13 | ssrin 3352 | . . . . . . . . 9 | |
14 | 8, 13 | ax-mp 5 | . . . . . . . 8 |
15 | nnord 4596 | . . . . . . . . 9 | |
16 | orddisj 4530 | . . . . . . . . 9 | |
17 | 15, 16 | syl 14 | . . . . . . . 8 |
18 | 14, 17 | sseqtrid 3197 | . . . . . . 7 |
19 | ss0 3455 | . . . . . . 7 | |
20 | 18, 19 | syl 14 | . . . . . 6 |
21 | 12, 20 | eqtrid 2215 | . . . . 5 |
22 | disjdif 3487 | . . . . 5 | |
23 | 21, 22 | jctil 310 | . . . 4 |
24 | unen 6794 | . . . 4 | |
25 | 11, 23, 24 | sylancr 412 | . . 3 |
26 | 25 | adantr 274 | . 2 |
27 | uncom 3271 | . . 3 | |
28 | nndifsnid 6486 | . . 3 | |
29 | 27, 28 | eqtrid 2215 | . 2 |
30 | phplem1 6830 | . 2 | |
31 | 26, 29, 30 | 3brtr3d 4020 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1348 wcel 2141 cvv 2730 cdif 3118 cun 3119 cin 3120 wss 3121 c0 3414 csn 3583 cop 3586 class class class wbr 3989 word 4347 csuc 4350 com 4574 wf1o 5197 cen 6716 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-en 6719 |
This theorem is referenced by: phplem3 6832 |
Copyright terms: Public domain | W3C validator |