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Theorem bj-nn0sucALT 13229
Description: Alternate proof of bj-nn0suc 13215, also constructive but from ax-inf2 13227, hence requiring ax-bdsetind 13219. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
bj-nn0sucALT (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
Distinct variable group:   𝑥,𝐴

Proof of Theorem bj-nn0sucALT
Dummy variables 𝑎 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-inf2 13227 . . 3 𝑎𝑦(𝑦𝑎 ↔ (𝑦 = ∅ ∨ ∃𝑧𝑎 𝑦 = suc 𝑧))
2 vex 2689 . . . . 5 𝑎 ∈ V
3 bdcv 13099 . . . . . 6 BOUNDED 𝑎
43bj-inf2vn 13225 . . . . 5 (𝑎 ∈ V → (∀𝑦(𝑦𝑎 ↔ (𝑦 = ∅ ∨ ∃𝑧𝑎 𝑦 = suc 𝑧)) → 𝑎 = ω))
52, 4ax-mp 5 . . . 4 (∀𝑦(𝑦𝑎 ↔ (𝑦 = ∅ ∨ ∃𝑧𝑎 𝑦 = suc 𝑧)) → 𝑎 = ω)
6 eleq2 2203 . . . . . . 7 (𝑎 = ω → (𝑦𝑎𝑦 ∈ ω))
7 rexeq 2627 . . . . . . . 8 (𝑎 = ω → (∃𝑧𝑎 𝑦 = suc 𝑧 ↔ ∃𝑧 ∈ ω 𝑦 = suc 𝑧))
87orbi2d 779 . . . . . . 7 (𝑎 = ω → ((𝑦 = ∅ ∨ ∃𝑧𝑎 𝑦 = suc 𝑧) ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧)))
96, 8bibi12d 234 . . . . . 6 (𝑎 = ω → ((𝑦𝑎 ↔ (𝑦 = ∅ ∨ ∃𝑧𝑎 𝑦 = suc 𝑧)) ↔ (𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧))))
109albidv 1796 . . . . 5 (𝑎 = ω → (∀𝑦(𝑦𝑎 ↔ (𝑦 = ∅ ∨ ∃𝑧𝑎 𝑦 = suc 𝑧)) ↔ ∀𝑦(𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧))))
11 nfcv 2281 . . . . . . . 8 𝑦𝐴
12 nfv 1508 . . . . . . . 8 𝑦(𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
13 eleq1 2202 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝑦 ∈ ω ↔ 𝐴 ∈ ω))
14 eqeq1 2146 . . . . . . . . . . 11 (𝑦 = 𝐴 → (𝑦 = ∅ ↔ 𝐴 = ∅))
15 suceq 4324 . . . . . . . . . . . . . 14 (𝑧 = 𝑥 → suc 𝑧 = suc 𝑥)
1615eqeq2d 2151 . . . . . . . . . . . . 13 (𝑧 = 𝑥 → (𝑦 = suc 𝑧𝑦 = suc 𝑥))
1716cbvrexv 2655 . . . . . . . . . . . 12 (∃𝑧 ∈ ω 𝑦 = suc 𝑧 ↔ ∃𝑥 ∈ ω 𝑦 = suc 𝑥)
18 eqeq1 2146 . . . . . . . . . . . . 13 (𝑦 = 𝐴 → (𝑦 = suc 𝑥𝐴 = suc 𝑥))
1918rexbidv 2438 . . . . . . . . . . . 12 (𝑦 = 𝐴 → (∃𝑥 ∈ ω 𝑦 = suc 𝑥 ↔ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
2017, 19syl5bb 191 . . . . . . . . . . 11 (𝑦 = 𝐴 → (∃𝑧 ∈ ω 𝑦 = suc 𝑧 ↔ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
2114, 20orbi12d 782 . . . . . . . . . 10 (𝑦 = 𝐴 → ((𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧) ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)))
2213, 21bibi12d 234 . . . . . . . . 9 (𝑦 = 𝐴 → ((𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧)) ↔ (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))))
23 bi1 117 . . . . . . . . 9 ((𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)) → (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)))
2422, 23syl6bi 162 . . . . . . . 8 (𝑦 = 𝐴 → ((𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧)) → (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))))
2511, 12, 24spcimgf 2766 . . . . . . 7 (𝐴 ∈ ω → (∀𝑦(𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧)) → (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))))
2625pm2.43b 52 . . . . . 6 (∀𝑦(𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧)) → (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)))
27 peano1 4508 . . . . . . . 8 ∅ ∈ ω
28 eleq1 2202 . . . . . . . 8 (𝐴 = ∅ → (𝐴 ∈ ω ↔ ∅ ∈ ω))
2927, 28mpbiri 167 . . . . . . 7 (𝐴 = ∅ → 𝐴 ∈ ω)
30 bj-peano2 13190 . . . . . . . . 9 (𝑥 ∈ ω → suc 𝑥 ∈ ω)
31 eleq1a 2211 . . . . . . . . . 10 (suc 𝑥 ∈ ω → (𝐴 = suc 𝑥𝐴 ∈ ω))
3231imp 123 . . . . . . . . 9 ((suc 𝑥 ∈ ω ∧ 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
3330, 32sylan 281 . . . . . . . 8 ((𝑥 ∈ ω ∧ 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
3433rexlimiva 2544 . . . . . . 7 (∃𝑥 ∈ ω 𝐴 = suc 𝑥𝐴 ∈ ω)
3529, 34jaoi 705 . . . . . 6 ((𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
3626, 35impbid1 141 . . . . 5 (∀𝑦(𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧)) → (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)))
3710, 36syl6bi 162 . . . 4 (𝑎 = ω → (∀𝑦(𝑦𝑎 ↔ (𝑦 = ∅ ∨ ∃𝑧𝑎 𝑦 = suc 𝑧)) → (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))))
385, 37mpcom 36 . . 3 (∀𝑦(𝑦𝑎 ↔ (𝑦 = ∅ ∨ ∃𝑧𝑎 𝑦 = suc 𝑧)) → (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)))
391, 38eximii 1581 . 2 𝑎(𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
40 bj-ex 13022 . 2 (∃𝑎(𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)) → (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)))
4139, 40ax-mp 5 1 (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wo 697  wal 1329   = wceq 1331  wex 1468  wcel 1480  wrex 2417  Vcvv 2686  c0 3363  suc csuc 4287  ωcom 4504
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-nul 4054  ax-pr 4131  ax-un 4355  ax-bd0 13064  ax-bdim 13065  ax-bdor 13067  ax-bdex 13070  ax-bdeq 13071  ax-bdel 13072  ax-bdsb 13073  ax-bdsep 13135  ax-bdsetind 13219  ax-inf2 13227
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-sn 3533  df-pr 3534  df-uni 3737  df-int 3772  df-suc 4293  df-iom 4505  df-bdc 13092  df-bj-ind 13178
This theorem is referenced by: (None)
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