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Theorem bj-nn0suc 15119
Description: Proof of (biconditional form of) nn0suc 4618 from the core axioms of CZF. See also bj-nn0sucALT 15133. 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 15105 . . 3 (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥𝐴 𝐴 = suc 𝑥))
2 bj-omtrans 15111 . . . . 5 (𝐴 ∈ ω → 𝐴 ⊆ ω)
3 ssrexv 3235 . . . . 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 789 . . 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 4608 . . . 4 ∅ ∈ ω
8 eleq1 2252 . . . 4 (𝐴 = ∅ → (𝐴 ∈ ω ↔ ∅ ∈ ω))
97, 8mpbiri 168 . . 3 (𝐴 = ∅ → 𝐴 ∈ ω)
10 bj-peano2 15094 . . . . 5 (𝑥 ∈ ω → suc 𝑥 ∈ ω)
11 eleq1a 2261 . . . . . 6 (suc 𝑥 ∈ ω → (𝐴 = suc 𝑥𝐴 ∈ ω))
1211imp 124 . . . . 5 ((suc 𝑥 ∈ ω ∧ 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
1310, 12sylan 283 . . . 4 ((𝑥 ∈ ω ∧ 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
1413rexlimiva 2602 . . 3 (∃𝑥 ∈ ω 𝐴 = suc 𝑥𝐴 ∈ ω)
159, 14jaoi 717 . 2 ((𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
166, 15impbii 126 1 (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wo 709   = wceq 1364  wcel 2160  wrex 2469  wss 3144  c0 3437  suc csuc 4380  ωcom 4604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-nul 4144  ax-pr 4224  ax-un 4448  ax-bd0 14968  ax-bdim 14969  ax-bdan 14970  ax-bdor 14971  ax-bdn 14972  ax-bdal 14973  ax-bdex 14974  ax-bdeq 14975  ax-bdel 14976  ax-bdsb 14977  ax-bdsep 15039  ax-infvn 15096
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-sn 3613  df-pr 3614  df-uni 3825  df-int 3860  df-suc 4386  df-iom 4605  df-bdc 14996  df-bj-ind 15082
This theorem is referenced by:  bj-findis  15134
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