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Mirrors > Home > NFE Home > Th. List > peano4 | GIF version |
Description: The successor operation is one-to-one over the finite cardinals. Theorem X.1.66 of [Rosser] p. 537. (Contributed by SF, 20-Jan-2015.) |
Ref | Expression |
---|---|
peano4 | ⊢ ((M ∈ Nn ∧ N ∈ Nn ∧ (M +c 1c) = (N +c 1c)) → M = N) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3simpa 952 | . 2 ⊢ ((M ∈ Nn ∧ N ∈ Nn ∧ (M +c 1c) = (N +c 1c)) → (M ∈ Nn ∧ N ∈ Nn )) | |
2 | simp3 957 | . 2 ⊢ ((M ∈ Nn ∧ N ∈ Nn ∧ (M +c 1c) = (N +c 1c)) → (M +c 1c) = (N +c 1c)) | |
3 | peano2 4403 | . . . 4 ⊢ (M ∈ Nn → (M +c 1c) ∈ Nn ) | |
4 | nulnnn 4556 | . . . . . 6 ⊢ ¬ ∅ ∈ Nn | |
5 | eleq1 2413 | . . . . . 6 ⊢ ((M +c 1c) = ∅ → ((M +c 1c) ∈ Nn ↔ ∅ ∈ Nn )) | |
6 | 4, 5 | mtbiri 294 | . . . . 5 ⊢ ((M +c 1c) = ∅ → ¬ (M +c 1c) ∈ Nn ) |
7 | 6 | necon2ai 2561 | . . . 4 ⊢ ((M +c 1c) ∈ Nn → (M +c 1c) ≠ ∅) |
8 | 3, 7 | syl 15 | . . 3 ⊢ (M ∈ Nn → (M +c 1c) ≠ ∅) |
9 | 8 | 3ad2ant1 976 | . 2 ⊢ ((M ∈ Nn ∧ N ∈ Nn ∧ (M +c 1c) = (N +c 1c)) → (M +c 1c) ≠ ∅) |
10 | prepeano4 4451 | . 2 ⊢ (((M ∈ Nn ∧ N ∈ Nn ) ∧ ((M +c 1c) = (N +c 1c) ∧ (M +c 1c) ≠ ∅)) → M = N) | |
11 | 1, 2, 9, 10 | syl12anc 1180 | 1 ⊢ ((M ∈ Nn ∧ N ∈ Nn ∧ (M +c 1c) = (N +c 1c)) → M = N) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 ∧ w3a 934 = wceq 1642 ∈ wcel 1710 ≠ wne 2516 ∅c0 3550 1cc1c 4134 Nn cnnc 4373 +c cplc 4375 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 |
This theorem is referenced by: suc11nnc 4558 phi11lem1 4595 fnfreclem3 6319 |
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