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Theorem peano2 4318
 Description: The successor of any natural number is a natural number. One of Peano's five postulates for arithmetic. Proposition 7.30(2) of [TakeutiZaring] p. 42. (Contributed by NM, 3-Sep-2003.)
Assertion
Ref Expression
peano2 (𝐴 ∈ ω → suc 𝐴 ∈ ω)

Proof of Theorem peano2
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2566 . 2 (𝐴 ∈ ω → 𝐴 ∈ V)
2 simpl 102 . . . . . 6 ((𝐴 ∈ V ∧ 𝑧 ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)}) → 𝐴 ∈ V)
3 eleq1 2100 . . . . . . . 8 (𝑥 = 𝐴 → (𝑥𝑧𝐴𝑧))
4 suceq 4139 . . . . . . . . 9 (𝑥 = 𝐴 → suc 𝑥 = suc 𝐴)
54eleq1d 2106 . . . . . . . 8 (𝑥 = 𝐴 → (suc 𝑥𝑧 ↔ suc 𝐴𝑧))
63, 5imbi12d 223 . . . . . . 7 (𝑥 = 𝐴 → ((𝑥𝑧 → suc 𝑥𝑧) ↔ (𝐴𝑧 → suc 𝐴𝑧)))
76adantl 262 . . . . . 6 (((𝐴 ∈ V ∧ 𝑧 ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)}) ∧ 𝑥 = 𝐴) → ((𝑥𝑧 → suc 𝑥𝑧) ↔ (𝐴𝑧 → suc 𝐴𝑧)))
8 df-clab 2027 . . . . . . . . 9 (𝑧 ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)} ↔ [𝑧 / 𝑦](∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦))
9 simpr 103 . . . . . . . . . . . 12 ((∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦) → ∀𝑥𝑦 suc 𝑥𝑦)
10 df-ral 2311 . . . . . . . . . . . 12 (∀𝑥𝑦 suc 𝑥𝑦 ↔ ∀𝑥(𝑥𝑦 → suc 𝑥𝑦))
119, 10sylib 127 . . . . . . . . . . 11 ((∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦) → ∀𝑥(𝑥𝑦 → suc 𝑥𝑦))
1211sbimi 1647 . . . . . . . . . 10 ([𝑧 / 𝑦](∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦) → [𝑧 / 𝑦]∀𝑥(𝑥𝑦 → suc 𝑥𝑦))
13 sbim 1827 . . . . . . . . . . . 12 ([𝑧 / 𝑦](𝑥𝑦 → suc 𝑥𝑦) ↔ ([𝑧 / 𝑦]𝑥𝑦 → [𝑧 / 𝑦]suc 𝑥𝑦))
14 elsb4 1853 . . . . . . . . . . . . 13 ([𝑧 / 𝑦]𝑥𝑦𝑥𝑧)
15 clelsb4 2143 . . . . . . . . . . . . 13 ([𝑧 / 𝑦]suc 𝑥𝑦 ↔ suc 𝑥𝑧)
1614, 15imbi12i 228 . . . . . . . . . . . 12 (([𝑧 / 𝑦]𝑥𝑦 → [𝑧 / 𝑦]suc 𝑥𝑦) ↔ (𝑥𝑧 → suc 𝑥𝑧))
1713, 16bitri 173 . . . . . . . . . . 11 ([𝑧 / 𝑦](𝑥𝑦 → suc 𝑥𝑦) ↔ (𝑥𝑧 → suc 𝑥𝑧))
1817sbalv 1881 . . . . . . . . . 10 ([𝑧 / 𝑦]∀𝑥(𝑥𝑦 → suc 𝑥𝑦) ↔ ∀𝑥(𝑥𝑧 → suc 𝑥𝑧))
1912, 18sylib 127 . . . . . . . . 9 ([𝑧 / 𝑦](∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦) → ∀𝑥(𝑥𝑧 → suc 𝑥𝑧))
208, 19sylbi 114 . . . . . . . 8 (𝑧 ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)} → ∀𝑥(𝑥𝑧 → suc 𝑥𝑧))
212019.21bi 1450 . . . . . . 7 (𝑧 ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)} → (𝑥𝑧 → suc 𝑥𝑧))
2221adantl 262 . . . . . 6 ((𝐴 ∈ V ∧ 𝑧 ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)}) → (𝑥𝑧 → suc 𝑥𝑧))
23 nfv 1421 . . . . . . 7 𝑥 𝐴 ∈ V
24 nfv 1421 . . . . . . . . 9 𝑥∅ ∈ 𝑦
25 nfra1 2355 . . . . . . . . 9 𝑥𝑥𝑦 suc 𝑥𝑦
2624, 25nfan 1457 . . . . . . . 8 𝑥(∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)
2726nfsab 2032 . . . . . . 7 𝑥 𝑧 ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)}
2823, 27nfan 1457 . . . . . 6 𝑥(𝐴 ∈ V ∧ 𝑧 ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)})
29 nfcvd 2179 . . . . . 6 ((𝐴 ∈ V ∧ 𝑧 ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)}) → 𝑥𝐴)
30 nfvd 1422 . . . . . 6 ((𝐴 ∈ V ∧ 𝑧 ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)}) → Ⅎ𝑥(𝐴𝑧 → suc 𝐴𝑧))
312, 7, 22, 28, 29, 30vtocldf 2605 . . . . 5 ((𝐴 ∈ V ∧ 𝑧 ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)}) → (𝐴𝑧 → suc 𝐴𝑧))
3231ralrimiva 2392 . . . 4 (𝐴 ∈ V → ∀𝑧 ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)} (𝐴𝑧 → suc 𝐴𝑧))
33 ralim 2380 . . . . 5 (∀𝑧 ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)} (𝐴𝑧 → suc 𝐴𝑧) → (∀𝑧 ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)}𝐴𝑧 → ∀𝑧 ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)}suc 𝐴𝑧))
34 elintg 3623 . . . . . 6 (𝐴 ∈ V → (𝐴 {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)} ↔ ∀𝑧 ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)}𝐴𝑧))
35 sucexg 4224 . . . . . . 7 (𝐴 ∈ V → suc 𝐴 ∈ V)
36 elintg 3623 . . . . . . 7 (suc 𝐴 ∈ V → (suc 𝐴 {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)} ↔ ∀𝑧 ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)}suc 𝐴𝑧))
3735, 36syl 14 . . . . . 6 (𝐴 ∈ V → (suc 𝐴 {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)} ↔ ∀𝑧 ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)}suc 𝐴𝑧))
3834, 37imbi12d 223 . . . . 5 (𝐴 ∈ V → ((𝐴 {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)} → suc 𝐴 {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)}) ↔ (∀𝑧 ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)}𝐴𝑧 → ∀𝑧 ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)}suc 𝐴𝑧)))
3933, 38syl5ibr 145 . . . 4 (𝐴 ∈ V → (∀𝑧 ∈ {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)} (𝐴𝑧 → suc 𝐴𝑧) → (𝐴 {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)} → suc 𝐴 {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)})))
4032, 39mpd 13 . . 3 (𝐴 ∈ V → (𝐴 {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)} → suc 𝐴 {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)}))
41 dfom3 4315 . . . 4 ω = {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)}
4241eleq2i 2104 . . 3 (𝐴 ∈ ω ↔ 𝐴 {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)})
4341eleq2i 2104 . . 3 (suc 𝐴 ∈ ω ↔ suc 𝐴 {𝑦 ∣ (∅ ∈ 𝑦 ∧ ∀𝑥𝑦 suc 𝑥𝑦)})
4440, 42, 433imtr4g 194 . 2 (𝐴 ∈ V → (𝐴 ∈ ω → suc 𝐴 ∈ ω))
451, 44mpcom 32 1 (𝐴 ∈ ω → suc 𝐴 ∈ ω)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1241   = wceq 1243   ∈ wcel 1393  [wsb 1645  {cab 2026  ∀wral 2306  Vcvv 2557  ∅c0 3224  ∩ cint 3615  suc csuc 4102  ωcom 4313 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-uni 3581  df-int 3616  df-suc 4108  df-iom 4314 This theorem is referenced by:  peano5  4321  limom  4336  peano2b  4337  nnregexmid  4342  frecsuclem1  5987  frecsuclem3  5990  frecrdg  5992  nnacl  6059  nnacom  6063  nnmsucr  6067  nnsucsssuc  6071  nnaword  6084  1onn  6093  2onn  6094  3onn  6095  4onn  6096  nnaordex  6100  php5  6321  phplem4dom  6324  php5dom  6325  phplem4on  6329  dif1en  6337  findcard  6345  findcard2  6346  findcard2s  6347  frec2uzrand  9191  frecuzrdgsuc  9201  frecfzennn  9203
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