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Theorem bj-nn0sucALT 13973
Description: Alternate proof of bj-nn0suc 13959, also constructive but from ax-inf2 13971, hence requiring ax-bdsetind 13963. (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 13971 . . 3 𝑎𝑦(𝑦𝑎 ↔ (𝑦 = ∅ ∨ ∃𝑧𝑎 𝑦 = suc 𝑧))
2 vex 2733 . . . . 5 𝑎 ∈ V
3 bdcv 13843 . . . . . 6 BOUNDED 𝑎
43bj-inf2vn 13969 . . . . 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 2234 . . . . . . 7 (𝑎 = ω → (𝑦𝑎𝑦 ∈ ω))
7 rexeq 2666 . . . . . . . 8 (𝑎 = ω → (∃𝑧𝑎 𝑦 = suc 𝑧 ↔ ∃𝑧 ∈ ω 𝑦 = suc 𝑧))
87orbi2d 785 . . . . . . 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 1817 . . . . 5 (𝑎 = ω → (∀𝑦(𝑦𝑎 ↔ (𝑦 = ∅ ∨ ∃𝑧𝑎 𝑦 = suc 𝑧)) ↔ ∀𝑦(𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧))))
11 nfcv 2312 . . . . . . . 8 𝑦𝐴
12 nfv 1521 . . . . . . . 8 𝑦(𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
13 eleq1 2233 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝑦 ∈ ω ↔ 𝐴 ∈ ω))
14 eqeq1 2177 . . . . . . . . . . 11 (𝑦 = 𝐴 → (𝑦 = ∅ ↔ 𝐴 = ∅))
15 suceq 4385 . . . . . . . . . . . . . 14 (𝑧 = 𝑥 → suc 𝑧 = suc 𝑥)
1615eqeq2d 2182 . . . . . . . . . . . . 13 (𝑧 = 𝑥 → (𝑦 = suc 𝑧𝑦 = suc 𝑥))
1716cbvrexv 2697 . . . . . . . . . . . 12 (∃𝑧 ∈ ω 𝑦 = suc 𝑧 ↔ ∃𝑥 ∈ ω 𝑦 = suc 𝑥)
18 eqeq1 2177 . . . . . . . . . . . . 13 (𝑦 = 𝐴 → (𝑦 = suc 𝑥𝐴 = suc 𝑥))
1918rexbidv 2471 . . . . . . . . . . . 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 788 . . . . . . . . . 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 biimp 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 2810 . . . . . . 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 4576 . . . . . . . 8 ∅ ∈ ω
28 eleq1 2233 . . . . . . . 8 (𝐴 = ∅ → (𝐴 ∈ ω ↔ ∅ ∈ ω))
2927, 28mpbiri 167 . . . . . . 7 (𝐴 = ∅ → 𝐴 ∈ ω)
30 bj-peano2 13934 . . . . . . . . 9 (𝑥 ∈ ω → suc 𝑥 ∈ ω)
31 eleq1a 2242 . . . . . . . . . 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 2582 . . . . . . 7 (∃𝑥 ∈ ω 𝐴 = suc 𝑥𝐴 ∈ ω)
3529, 34jaoi 711 . . . . . 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 1595 . 2 𝑎(𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
40 bj-ex 13756 . 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 703  wal 1346   = wceq 1348  wex 1485  wcel 2141  wrex 2449  Vcvv 2730  c0 3414  suc csuc 4348  ωcom 4572
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-nul 4113  ax-pr 4192  ax-un 4416  ax-bd0 13808  ax-bdim 13809  ax-bdor 13811  ax-bdex 13814  ax-bdeq 13815  ax-bdel 13816  ax-bdsb 13817  ax-bdsep 13879  ax-bdsetind 13963  ax-inf2 13971
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-sn 3587  df-pr 3588  df-uni 3795  df-int 3830  df-suc 4354  df-iom 4573  df-bdc 13836  df-bj-ind 13922
This theorem is referenced by: (None)
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