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Description: A natural number is not equinumerous to its successor. Corollary 10.21(1) of [TakeutiZaring] p. 90. (Contributed by NM, 26-Jul-2004.) |
Ref | Expression |
---|---|
php5 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 | . . . 4 | |
2 | suceq 4364 | . . . 4 | |
3 | 1, 2 | breq12d 3980 | . . 3 |
4 | 3 | notbid 657 | . 2 |
5 | id 19 | . . . 4 | |
6 | suceq 4364 | . . . 4 | |
7 | 5, 6 | breq12d 3980 | . . 3 |
8 | 7 | notbid 657 | . 2 |
9 | id 19 | . . . 4 | |
10 | suceq 4364 | . . . 4 | |
11 | 9, 10 | breq12d 3980 | . . 3 |
12 | 11 | notbid 657 | . 2 |
13 | id 19 | . . . 4 | |
14 | suceq 4364 | . . . 4 | |
15 | 13, 14 | breq12d 3980 | . . 3 |
16 | 15 | notbid 657 | . 2 |
17 | peano1 4555 | . . . . 5 | |
18 | peano3 4557 | . . . . 5 | |
19 | 17, 18 | ax-mp 5 | . . . 4 |
20 | en0 6742 | . . . 4 | |
21 | 19, 20 | nemtbir 2416 | . . 3 |
22 | ensymb 6727 | . . 3 | |
23 | 21, 22 | mtbi 660 | . 2 |
24 | peano2 4556 | . . . 4 | |
25 | vex 2715 | . . . . 5 | |
26 | 25 | sucex 4460 | . . . . 5 |
27 | 25, 26 | phplem4 6802 | . . . 4 |
28 | 24, 27 | mpdan 418 | . . 3 |
29 | 28 | con3d 621 | . 2 |
30 | 4, 8, 12, 16, 23, 29 | finds 4561 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wceq 1335 wcel 2128 wne 2327 c0 3395 class class class wbr 3967 csuc 4327 com 4551 cen 6685 |
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 4084 ax-nul 4092 ax-pow 4137 ax-pr 4171 ax-un 4395 ax-setind 4498 ax-iinf 4549 |
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-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3396 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-br 3968 df-opab 4028 df-tr 4065 df-id 4255 df-iord 4328 df-on 4330 df-suc 4333 df-iom 4552 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 df-res 4600 df-ima 4601 df-iota 5137 df-fun 5174 df-fn 5175 df-f 5176 df-f1 5177 df-fo 5178 df-f1o 5179 df-fv 5180 df-er 6482 df-en 6688 |
This theorem is referenced by: snnen2og 6806 1nen2 6808 php5dom 6810 php5fin 6829 |
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