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Theorem bj-nn0suc 11505
Description: Proof of (biconditional form of) nn0suc 4409 from the core axioms of CZF. See also bj-nn0sucALT 11519. As a characterization of the elements of ω, this could be labeled "elom". (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-nn0suc (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
Distinct variable group:   𝑥,𝐴

Proof of Theorem bj-nn0suc
StepHypRef Expression
1 bj-nn0suc0 11491 . . 3 (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥𝐴 𝐴 = suc 𝑥))
2 bj-omtrans 11497 . . . . 5 (𝐴 ∈ ω → 𝐴 ⊆ ω)
3 ssrexv 3084 . . . . 5 (𝐴 ⊆ ω → (∃𝑥𝐴 𝐴 = suc 𝑥 → ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
42, 3syl 14 . . . 4 (𝐴 ∈ ω → (∃𝑥𝐴 𝐴 = suc 𝑥 → ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
54orim2d 737 . . 3 (𝐴 ∈ ω → ((𝐴 = ∅ ∨ ∃𝑥𝐴 𝐴 = suc 𝑥) → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)))
61, 5mpd 13 . 2 (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
7 peano1 4399 . . . 4 ∅ ∈ ω
8 eleq1 2150 . . . 4 (𝐴 = ∅ → (𝐴 ∈ ω ↔ ∅ ∈ ω))
97, 8mpbiri 166 . . 3 (𝐴 = ∅ → 𝐴 ∈ ω)
10 bj-peano2 11480 . . . . 5 (𝑥 ∈ ω → suc 𝑥 ∈ ω)
11 eleq1a 2159 . . . . . 6 (suc 𝑥 ∈ ω → (𝐴 = suc 𝑥𝐴 ∈ ω))
1211imp 122 . . . . 5 ((suc 𝑥 ∈ ω ∧ 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
1310, 12sylan 277 . . . 4 ((𝑥 ∈ ω ∧ 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
1413rexlimiva 2484 . . 3 (∃𝑥 ∈ ω 𝐴 = suc 𝑥𝐴 ∈ ω)
159, 14jaoi 671 . 2 ((𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
166, 15impbii 124 1 (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wo 664   = wceq 1289  wcel 1438  wrex 2360  wss 2997  c0 3284  suc csuc 4183  ωcom 4395
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-nul 3957  ax-pr 4027  ax-un 4251  ax-bd0 11350  ax-bdim 11351  ax-bdan 11352  ax-bdor 11353  ax-bdn 11354  ax-bdal 11355  ax-bdex 11356  ax-bdeq 11357  ax-bdel 11358  ax-bdsb 11359  ax-bdsep 11421  ax-infvn 11482
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-sn 3447  df-pr 3448  df-uni 3649  df-int 3684  df-suc 4189  df-iom 4396  df-bdc 11378  df-bj-ind 11468
This theorem is referenced by:  bj-findis  11520
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