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Theorem bj-nn0sucALT 11216
Description: Alternate proof of bj-nn0suc 11202, also constructive but from ax-inf2 11214, hence requiring ax-bdsetind 11206. (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 11214 . . 3 𝑎𝑦(𝑦𝑎 ↔ (𝑦 = ∅ ∨ ∃𝑧𝑎 𝑦 = suc 𝑧))
2 vex 2615 . . . . 5 𝑎 ∈ V
3 bdcv 11082 . . . . . 6 BOUNDED 𝑎
43bj-inf2vn 11212 . . . . 5 (𝑎 ∈ V → (∀𝑦(𝑦𝑎 ↔ (𝑦 = ∅ ∨ ∃𝑧𝑎 𝑦 = suc 𝑧)) → 𝑎 = ω))
52, 4ax-mp 7 . . . 4 (∀𝑦(𝑦𝑎 ↔ (𝑦 = ∅ ∨ ∃𝑧𝑎 𝑦 = suc 𝑧)) → 𝑎 = ω)
6 eleq2 2146 . . . . . . 7 (𝑎 = ω → (𝑦𝑎𝑦 ∈ ω))
7 rexeq 2556 . . . . . . . 8 (𝑎 = ω → (∃𝑧𝑎 𝑦 = suc 𝑧 ↔ ∃𝑧 ∈ ω 𝑦 = suc 𝑧))
87orbi2d 737 . . . . . . 7 (𝑎 = ω → ((𝑦 = ∅ ∨ ∃𝑧𝑎 𝑦 = suc 𝑧) ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧)))
96, 8bibi12d 233 . . . . . 6 (𝑎 = ω → ((𝑦𝑎 ↔ (𝑦 = ∅ ∨ ∃𝑧𝑎 𝑦 = suc 𝑧)) ↔ (𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧))))
109albidv 1747 . . . . 5 (𝑎 = ω → (∀𝑦(𝑦𝑎 ↔ (𝑦 = ∅ ∨ ∃𝑧𝑎 𝑦 = suc 𝑧)) ↔ ∀𝑦(𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧))))
11 nfcv 2223 . . . . . . . 8 𝑦𝐴
12 nfv 1462 . . . . . . . 8 𝑦(𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
13 eleq1 2145 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝑦 ∈ ω ↔ 𝐴 ∈ ω))
14 eqeq1 2089 . . . . . . . . . . 11 (𝑦 = 𝐴 → (𝑦 = ∅ ↔ 𝐴 = ∅))
15 suceq 4193 . . . . . . . . . . . . . 14 (𝑧 = 𝑥 → suc 𝑧 = suc 𝑥)
1615eqeq2d 2094 . . . . . . . . . . . . 13 (𝑧 = 𝑥 → (𝑦 = suc 𝑧𝑦 = suc 𝑥))
1716cbvrexv 2584 . . . . . . . . . . . 12 (∃𝑧 ∈ ω 𝑦 = suc 𝑧 ↔ ∃𝑥 ∈ ω 𝑦 = suc 𝑥)
18 eqeq1 2089 . . . . . . . . . . . . 13 (𝑦 = 𝐴 → (𝑦 = suc 𝑥𝐴 = suc 𝑥))
1918rexbidv 2375 . . . . . . . . . . . 12 (𝑦 = 𝐴 → (∃𝑥 ∈ ω 𝑦 = suc 𝑥 ↔ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
2017, 19syl5bb 190 . . . . . . . . . . 11 (𝑦 = 𝐴 → (∃𝑧 ∈ ω 𝑦 = suc 𝑧 ↔ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
2114, 20orbi12d 740 . . . . . . . . . 10 (𝑦 = 𝐴 → ((𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧) ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)))
2213, 21bibi12d 233 . . . . . . . . 9 (𝑦 = 𝐴 → ((𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧)) ↔ (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))))
23 bi1 116 . . . . . . . . 9 ((𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)) → (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)))
2422, 23syl6bi 161 . . . . . . . 8 (𝑦 = 𝐴 → ((𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧)) → (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))))
2511, 12, 24spcimgf 2689 . . . . . . 7 (𝐴 ∈ ω → (∀𝑦(𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧)) → (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))))
2625pm2.43b 51 . . . . . 6 (∀𝑦(𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧)) → (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)))
27 peano1 4372 . . . . . . . 8 ∅ ∈ ω
28 eleq1 2145 . . . . . . . 8 (𝐴 = ∅ → (𝐴 ∈ ω ↔ ∅ ∈ ω))
2927, 28mpbiri 166 . . . . . . 7 (𝐴 = ∅ → 𝐴 ∈ ω)
30 bj-peano2 11177 . . . . . . . . 9 (𝑥 ∈ ω → suc 𝑥 ∈ ω)
31 eleq1a 2154 . . . . . . . . . 10 (suc 𝑥 ∈ ω → (𝐴 = suc 𝑥𝐴 ∈ ω))
3231imp 122 . . . . . . . . 9 ((suc 𝑥 ∈ ω ∧ 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
3330, 32sylan 277 . . . . . . . 8 ((𝑥 ∈ ω ∧ 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
3433rexlimiva 2478 . . . . . . 7 (∃𝑥 ∈ ω 𝐴 = suc 𝑥𝐴 ∈ ω)
3529, 34jaoi 669 . . . . . 6 ((𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
3626, 35impbid1 140 . . . . 5 (∀𝑦(𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧)) → (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)))
3710, 36syl6bi 161 . . . 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 1534 . 2 𝑎(𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
40 bj-ex 11006 . 2 (∃𝑎(𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)) → (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)))
4139, 40ax-mp 7 1 (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wo 662  wal 1283   = wceq 1285  wex 1422  wcel 1434  wrex 2354  Vcvv 2612  c0 3269  suc csuc 4156  ωcom 4368
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-nul 3930  ax-pr 4000  ax-un 4224  ax-bd0 11047  ax-bdim 11048  ax-bdor 11050  ax-bdex 11053  ax-bdeq 11054  ax-bdel 11055  ax-bdsb 11056  ax-bdsep 11118  ax-bdsetind 11206  ax-inf2 11214
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2614  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-sn 3428  df-pr 3429  df-uni 3628  df-int 3663  df-suc 4162  df-iom 4369  df-bdc 11075  df-bj-ind 11165
This theorem is referenced by: (None)
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