Theorem bj-nn0suc 13162

Description: Proof of (biconditional form of) nn0suc 4518 from the core axioms of CZF. See also bj-nn0sucALT 13176. 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

Step	Hyp	Ref	Expression
1		bj-nn0suc0 13148	. . 3 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ 𝐴 𝐴 = suc 𝑥))
2		bj-omtrans 13154	. . . . 5 ⊢ (𝐴 ∈ ω → 𝐴 ⊆ ω)
3		ssrexv 3162	. . . . 5 ⊢ (𝐴 ⊆ ω → (∃𝑥 ∈ 𝐴 𝐴 = suc 𝑥 → ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
4	2, 3	syl 14	. . . 4 ⊢ (𝐴 ∈ ω → (∃𝑥 ∈ 𝐴 𝐴 = suc 𝑥 → ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
5	4	orim2d 777	. . 3 ⊢ (𝐴 ∈ ω → ((𝐴 = ∅ ∨ ∃𝑥 ∈ 𝐴 𝐴 = suc 𝑥) → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥)))
6	1, 5	mpd 13	. 2 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))
7		peano1 4508	. . . 4 ⊢ ∅ ∈ ω
8		eleq1 2202	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 ∈ ω ↔ ∅ ∈ ω))
9	7, 8	mpbiri 167	. . 3 ⊢ (𝐴 = ∅ → 𝐴 ∈ ω)
10		bj-peano2 13137	. . . . 5 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
11		eleq1a 2211	. . . . . 6 ⊢ (suc 𝑥 ∈ ω → (𝐴 = suc 𝑥 → 𝐴 ∈ ω))
12	11	imp 123	. . . . 5 ⊢ ((suc 𝑥 ∈ ω ∧ 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
13	10, 12	sylan 281	. . . 4 ⊢ ((𝑥 ∈ ω ∧ 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
14	13	rexlimiva 2544	. . 3 ⊢ (∃𝑥 ∈ ω 𝐴 = suc 𝑥 → 𝐴 ∈ ω)
15	9, 14	jaoi 705	. 2 ⊢ ((𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
16	6, 15	impbii 125	1 ⊢ (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))

Syntax hints:   → wi 4   ↔ wb 104   ∨ wo 697   = wceq 1331   ∈ wcel 1480  ∃wrex 2417   ⊆ wss 3071  ∅c0 3363  suc csuc 4287  ωcom 4504

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-nul 4054  ax-pr 4131  ax-un 4355  ax-bd0 13011  ax-bdim 13012  ax-bdan 13013  ax-bdor 13014  ax-bdn 13015  ax-bdal 13016  ax-bdex 13017  ax-bdeq 13018  ax-bdel 13019  ax-bdsb 13020  ax-bdsep 13082  ax-infvn 13139

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-sn 3533  df-pr 3534  df-uni 3737  df-int 3772  df-suc 4293  df-iom 4505  df-bdc 13039  df-bj-ind 13125

This theorem is referenced by:  bj-findis  13177