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Theorem bj-nn0suc 14956
Description: Proof of (biconditional form of) nn0suc 4615 from the core axioms of CZF. See also bj-nn0sucALT 14970. 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 14942 . . 3 (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥𝐴 𝐴 = suc 𝑥))
2 bj-omtrans 14948 . . . . 5 (𝐴 ∈ ω → 𝐴 ⊆ ω)
3 ssrexv 3232 . . . . 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 4605 . . . 4 ∅ ∈ ω
8 eleq1 2250 . . . 4 (𝐴 = ∅ → (𝐴 ∈ ω ↔ ∅ ∈ ω))
97, 8mpbiri 168 . . 3 (𝐴 = ∅ → 𝐴 ∈ ω)
10 bj-peano2 14931 . . . . 5 (𝑥 ∈ ω → suc 𝑥 ∈ ω)
11 eleq1a 2259 . . . . . 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 2599 . . 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 1363  wcel 2158  wrex 2466  wss 3141  c0 3434  suc csuc 4377  ωcom 4601
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-nul 4141  ax-pr 4221  ax-un 4445  ax-bd0 14805  ax-bdim 14806  ax-bdan 14807  ax-bdor 14808  ax-bdn 14809  ax-bdal 14810  ax-bdex 14811  ax-bdeq 14812  ax-bdel 14813  ax-bdsb 14814  ax-bdsep 14876  ax-infvn 14933
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-sn 3610  df-pr 3611  df-uni 3822  df-int 3857  df-suc 4383  df-iom 4602  df-bdc 14833  df-bj-ind 14919
This theorem is referenced by:  bj-findis  14971
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