Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bj-nn0sucALT GIF version

Theorem bj-nn0sucALT 10476
Description: Alternate proof of bj-nn0suc 10462, also constructive but from ax-inf2 10474, hence requiring ax-bdsetind 10466. (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 10474 . . 3 𝑎𝑦(𝑦𝑎 ↔ (𝑦 = ∅ ∨ ∃𝑧𝑎 𝑦 = suc 𝑧))
2 vex 2577 . . . . 5 𝑎 ∈ V
3 bdcv 10341 . . . . . 6 BOUNDED 𝑎
43bj-inf2vn 10472 . . . . 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 2117 . . . . . . 7 (𝑎 = ω → (𝑦𝑎𝑦 ∈ ω))
7 rexeq 2523 . . . . . . . 8 (𝑎 = ω → (∃𝑧𝑎 𝑦 = suc 𝑧 ↔ ∃𝑧 ∈ ω 𝑦 = suc 𝑧))
87orbi2d 714 . . . . . . 7 (𝑎 = ω → ((𝑦 = ∅ ∨ ∃𝑧𝑎 𝑦 = suc 𝑧) ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧)))
96, 8bibi12d 228 . . . . . 6 (𝑎 = ω → ((𝑦𝑎 ↔ (𝑦 = ∅ ∨ ∃𝑧𝑎 𝑦 = suc 𝑧)) ↔ (𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧))))
109albidv 1721 . . . . 5 (𝑎 = ω → (∀𝑦(𝑦𝑎 ↔ (𝑦 = ∅ ∨ ∃𝑧𝑎 𝑦 = suc 𝑧)) ↔ ∀𝑦(𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧))))
11 nfcv 2194 . . . . . . . 8 𝑦𝐴
12 nfv 1437 . . . . . . . 8 𝑦(𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
13 eleq1 2116 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝑦 ∈ ω ↔ 𝐴 ∈ ω))
14 eqeq1 2062 . . . . . . . . . . 11 (𝑦 = 𝐴 → (𝑦 = ∅ ↔ 𝐴 = ∅))
15 suceq 4166 . . . . . . . . . . . . . 14 (𝑧 = 𝑥 → suc 𝑧 = suc 𝑥)
1615eqeq2d 2067 . . . . . . . . . . . . 13 (𝑧 = 𝑥 → (𝑦 = suc 𝑧𝑦 = suc 𝑥))
1716cbvrexv 2551 . . . . . . . . . . . 12 (∃𝑧 ∈ ω 𝑦 = suc 𝑧 ↔ ∃𝑥 ∈ ω 𝑦 = suc 𝑥)
18 eqeq1 2062 . . . . . . . . . . . . 13 (𝑦 = 𝐴 → (𝑦 = suc 𝑥𝐴 = suc 𝑥))
1918rexbidv 2344 . . . . . . . . . . . 12 (𝑦 = 𝐴 → (∃𝑥 ∈ ω 𝑦 = suc 𝑥 ↔ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
2017, 19syl5bb 185 . . . . . . . . . . 11 (𝑦 = 𝐴 → (∃𝑧 ∈ ω 𝑦 = suc 𝑧 ↔ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
2114, 20orbi12d 717 . . . . . . . . . 10 (𝑦 = 𝐴 → ((𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧) ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)))
2213, 21bibi12d 228 . . . . . . . . 9 (𝑦 = 𝐴 → ((𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧)) ↔ (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))))
23 bi1 115 . . . . . . . . 9 ((𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)) → (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)))
2422, 23syl6bi 156 . . . . . . . 8 (𝑦 = 𝐴 → ((𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧)) → (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))))
2511, 12, 24spcimgf 2650 . . . . . . 7 (𝐴 ∈ ω → (∀𝑦(𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧)) → (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))))
2625pm2.43b 50 . . . . . 6 (∀𝑦(𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧)) → (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)))
27 peano1 4344 . . . . . . . 8 ∅ ∈ ω
28 eleq1 2116 . . . . . . . 8 (𝐴 = ∅ → (𝐴 ∈ ω ↔ ∅ ∈ ω))
2927, 28mpbiri 161 . . . . . . 7 (𝐴 = ∅ → 𝐴 ∈ ω)
30 bj-peano2 10436 . . . . . . . . 9 (𝑥 ∈ ω → suc 𝑥 ∈ ω)
31 eleq1a 2125 . . . . . . . . . 10 (suc 𝑥 ∈ ω → (𝐴 = suc 𝑥𝐴 ∈ ω))
3231imp 119 . . . . . . . . 9 ((suc 𝑥 ∈ ω ∧ 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
3330, 32sylan 271 . . . . . . . 8 ((𝑥 ∈ ω ∧ 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
3433rexlimiva 2445 . . . . . . 7 (∃𝑥 ∈ ω 𝐴 = suc 𝑥𝐴 ∈ ω)
3529, 34jaoi 646 . . . . . 6 ((𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
3626, 35impbid1 134 . . . . 5 (∀𝑦(𝑦 ∈ ω ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ ω 𝑦 = suc 𝑧)) → (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)))
3710, 36syl6bi 156 . . . 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 1509 . 2 𝑎(𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
40 bj-ex 10275 . 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 102  wo 639  wal 1257   = wceq 1259  wex 1397  wcel 1409  wrex 2324  Vcvv 2574  c0 3251  suc csuc 4129  ωcom 4340
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-nul 3910  ax-pr 3971  ax-un 4197  ax-bd0 10306  ax-bdim 10307  ax-bdor 10309  ax-bdex 10312  ax-bdeq 10313  ax-bdel 10314  ax-bdsb 10315  ax-bdsep 10377  ax-bdsetind 10466  ax-inf2 10474
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-sn 3408  df-pr 3409  df-uni 3608  df-int 3643  df-suc 4135  df-iom 4341  df-bdc 10334  df-bj-ind 10424
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator