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Theorem bj-nn0sucALT 10103
Description: Alternate proof of bj-nn0suc 10089, also constructive but from ax-inf2 10101, hence requiring ax-bdsetind 10093. (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 10101 . . 3 𝑎𝑦(𝑦𝑎 ↔ (𝑦 = ∅ ∨ ∃𝑧𝑎 𝑦 = suc 𝑧))
2 vex 2560 . . . . 5 𝑎 ∈ V
3 bdcv 9968 . . . . . 6 BOUNDED 𝑎
43bj-inf2vn 10099 . . . . 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 2101 . . . . . . 7 (𝑎 = ω → (𝑦𝑎𝑦 ∈ ω))
7 rexeq 2506 . . . . . . . 8 (𝑎 = ω → (∃𝑧𝑎 𝑦 = suc 𝑧 ↔ ∃𝑧 ∈ ω 𝑦 = suc 𝑧))
87orbi2d 704 . . . . . . 7 (𝑎 = ω → ((𝑦 = ∅ ∨ ∃𝑧𝑎 𝑦 = suc 𝑧) ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧)))
96, 8bibi12d 224 . . . . . 6 (𝑎 = ω → ((𝑦𝑎 ↔ (𝑦 = ∅ ∨ ∃𝑧𝑎 𝑦 = suc 𝑧)) ↔ (𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧))))
109albidv 1705 . . . . 5 (𝑎 = ω → (∀𝑦(𝑦𝑎 ↔ (𝑦 = ∅ ∨ ∃𝑧𝑎 𝑦 = suc 𝑧)) ↔ ∀𝑦(𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧))))
11 nfcv 2178 . . . . . . . 8 𝑦𝐴
12 nfv 1421 . . . . . . . 8 𝑦(𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
13 eleq1 2100 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝑦 ∈ ω ↔ 𝐴 ∈ ω))
14 eqeq1 2046 . . . . . . . . . . 11 (𝑦 = 𝐴 → (𝑦 = ∅ ↔ 𝐴 = ∅))
15 suceq 4139 . . . . . . . . . . . . . 14 (𝑧 = 𝑥 → suc 𝑧 = suc 𝑥)
1615eqeq2d 2051 . . . . . . . . . . . . 13 (𝑧 = 𝑥 → (𝑦 = suc 𝑧𝑦 = suc 𝑥))
1716cbvrexv 2534 . . . . . . . . . . . 12 (∃𝑧 ∈ ω 𝑦 = suc 𝑧 ↔ ∃𝑥 ∈ ω 𝑦 = suc 𝑥)
18 eqeq1 2046 . . . . . . . . . . . . 13 (𝑦 = 𝐴 → (𝑦 = suc 𝑥𝐴 = suc 𝑥))
1918rexbidv 2327 . . . . . . . . . . . 12 (𝑦 = 𝐴 → (∃𝑥 ∈ ω 𝑦 = suc 𝑥 ↔ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
2017, 19syl5bb 181 . . . . . . . . . . 11 (𝑦 = 𝐴 → (∃𝑧 ∈ ω 𝑦 = suc 𝑧 ↔ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
2114, 20orbi12d 707 . . . . . . . . . 10 (𝑦 = 𝐴 → ((𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧) ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)))
2213, 21bibi12d 224 . . . . . . . . 9 (𝑦 = 𝐴 → ((𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧)) ↔ (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))))
23 bi1 111 . . . . . . . . 9 ((𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)) → (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)))
2422, 23syl6bi 152 . . . . . . . 8 (𝑦 = 𝐴 → ((𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧)) → (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))))
2511, 12, 24spcimgf 2633 . . . . . . 7 (𝐴 ∈ ω → (∀𝑦(𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧)) → (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))))
2625pm2.43b 46 . . . . . 6 (∀𝑦(𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧)) → (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)))
27 peano1 4317 . . . . . . . 8 ∅ ∈ ω
28 eleq1 2100 . . . . . . . 8 (𝐴 = ∅ → (𝐴 ∈ ω ↔ ∅ ∈ ω))
2927, 28mpbiri 157 . . . . . . 7 (𝐴 = ∅ → 𝐴 ∈ ω)
30 bj-peano2 10063 . . . . . . . . 9 (𝑥 ∈ ω → suc 𝑥 ∈ ω)
31 eleq1a 2109 . . . . . . . . . 10 (suc 𝑥 ∈ ω → (𝐴 = suc 𝑥𝐴 ∈ ω))
3231imp 115 . . . . . . . . 9 ((suc 𝑥 ∈ ω ∧ 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
3330, 32sylan 267 . . . . . . . 8 ((𝑥 ∈ ω ∧ 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
3433rexlimiva 2428 . . . . . . 7 (∃𝑥 ∈ ω 𝐴 = suc 𝑥𝐴 ∈ ω)
3529, 34jaoi 636 . . . . . 6 ((𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
3626, 35impbid1 130 . . . . 5 (∀𝑦(𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧)) → (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)))
3710, 36syl6bi 152 . . . 4 (𝑎 = ω → (∀𝑦(𝑦𝑎 ↔ (𝑦 = ∅ ∨ ∃𝑧𝑎 𝑦 = suc 𝑧)) → (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))))
385, 37mpcom 32 . . 3 (∀𝑦(𝑦𝑎 ↔ (𝑦 = ∅ ∨ ∃𝑧𝑎 𝑦 = suc 𝑧)) → (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)))
391, 38eximii 1493 . 2 𝑎(𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
40 bj-ex 9902 . 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 98  wo 629  wal 1241   = wceq 1243  wex 1381  wcel 1393  wrex 2307  Vcvv 2557  c0 3224  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-in1 544  ax-in2 545  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-nul 3883  ax-pr 3944  ax-un 4170  ax-bd0 9933  ax-bdim 9934  ax-bdor 9936  ax-bdex 9939  ax-bdeq 9940  ax-bdel 9941  ax-bdsb 9942  ax-bdsep 10004  ax-bdsetind 10093  ax-inf2 10101
This theorem depends on definitions:  df-bi 110  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-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-sn 3381  df-pr 3382  df-uni 3581  df-int 3616  df-suc 4108  df-iom 4314  df-bdc 9961  df-bj-ind 10051
This theorem is referenced by: (None)
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