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Theorem bj-nn0suc 12985
Description: Proof of (biconditional form of) nn0suc 4486 from the core axioms of CZF. See also bj-nn0sucALT 12999. 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 12971 . . 3 (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥𝐴 𝐴 = suc 𝑥))
2 bj-omtrans 12977 . . . . 5 (𝐴 ∈ ω → 𝐴 ⊆ ω)
3 ssrexv 3130 . . . . 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 760 . . 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 4476 . . . 4 ∅ ∈ ω
8 eleq1 2178 . . . 4 (𝐴 = ∅ → (𝐴 ∈ ω ↔ ∅ ∈ ω))
97, 8mpbiri 167 . . 3 (𝐴 = ∅ → 𝐴 ∈ ω)
10 bj-peano2 12960 . . . . 5 (𝑥 ∈ ω → suc 𝑥 ∈ ω)
11 eleq1a 2187 . . . . . 6 (suc 𝑥 ∈ ω → (𝐴 = suc 𝑥𝐴 ∈ ω))
1211imp 123 . . . . 5 ((suc 𝑥 ∈ ω ∧ 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
1310, 12sylan 279 . . . 4 ((𝑥 ∈ ω ∧ 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
1413rexlimiva 2519 . . 3 (∃𝑥 ∈ ω 𝐴 = suc 𝑥𝐴 ∈ ω)
159, 14jaoi 688 . 2 ((𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
166, 15impbii 125 1 (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wo 680   = wceq 1314  wcel 1463  wrex 2392  wss 3039  c0 3331  suc csuc 4255  ωcom 4472
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-nul 4022  ax-pr 4099  ax-un 4323  ax-bd0 12834  ax-bdim 12835  ax-bdan 12836  ax-bdor 12837  ax-bdn 12838  ax-bdal 12839  ax-bdex 12840  ax-bdeq 12841  ax-bdel 12842  ax-bdsb 12843  ax-bdsep 12905  ax-infvn 12962
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-sn 3501  df-pr 3502  df-uni 3705  df-int 3740  df-suc 4261  df-iom 4473  df-bdc 12862  df-bj-ind 12948
This theorem is referenced by:  bj-findis  13000
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