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Theorem bj-nn0suc 10476
Description: Proof of (biconditional form of) nn0suc 4355 from the core axioms of CZF. See also bj-nn0sucALT 10490. 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 10462 . . 3 (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥𝐴 𝐴 = suc 𝑥))
2 bj-omtrans 10468 . . . . 5 (𝐴 ∈ ω → 𝐴 ⊆ ω)
3 ssrexv 3033 . . . . 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 712 . . 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 4345 . . . 4 ∅ ∈ ω
8 eleq1 2116 . . . 4 (𝐴 = ∅ → (𝐴 ∈ ω ↔ ∅ ∈ ω))
97, 8mpbiri 161 . . 3 (𝐴 = ∅ → 𝐴 ∈ ω)
10 bj-peano2 10450 . . . . 5 (𝑥 ∈ ω → suc 𝑥 ∈ ω)
11 eleq1a 2125 . . . . . 6 (suc 𝑥 ∈ ω → (𝐴 = suc 𝑥𝐴 ∈ ω))
1211imp 119 . . . . 5 ((suc 𝑥 ∈ ω ∧ 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
1310, 12sylan 271 . . . 4 ((𝑥 ∈ ω ∧ 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
1413rexlimiva 2445 . . 3 (∃𝑥 ∈ ω 𝐴 = suc 𝑥𝐴 ∈ ω)
159, 14jaoi 646 . 2 ((𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
166, 15impbii 121 1 (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wo 639   = wceq 1259  wcel 1409  wrex 2324  wss 2945  c0 3252  suc csuc 4130  ωcom 4341
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 3911  ax-pr 3972  ax-un 4198  ax-bd0 10320  ax-bdim 10321  ax-bdan 10322  ax-bdor 10323  ax-bdn 10324  ax-bdal 10325  ax-bdex 10326  ax-bdeq 10327  ax-bdel 10328  ax-bdsb 10329  ax-bdsep 10391  ax-infvn 10453
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265  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 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-sn 3409  df-pr 3410  df-uni 3609  df-int 3644  df-suc 4136  df-iom 4342  df-bdc 10348  df-bj-ind 10438
This theorem is referenced by:  bj-findis  10491
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