Theorem bj-nn0suc 15764

Description: Proof of (biconditional form of) nn0suc 4650 from the core axioms of CZF. See also bj-nn0sucALT 15778. 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 15750	. . 3 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ 𝐴 𝐴 = suc 𝑥))
2		bj-omtrans 15756	. . . . 5 ⊢ (𝐴 ∈ ω → 𝐴 ⊆ ω)
3		ssrexv 3257	. . . . 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 789	. . 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 4640	. . . 4 ⊢ ∅ ∈ ω
8		eleq1 2267	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 ∈ ω ↔ ∅ ∈ ω))
9	7, 8	mpbiri 168	. . 3 ⊢ (𝐴 = ∅ → 𝐴 ∈ ω)
10		bj-peano2 15739	. . . . 5 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
11		eleq1a 2276	. . . . . 6 ⊢ (suc 𝑥 ∈ ω → (𝐴 = suc 𝑥 → 𝐴 ∈ ω))
12	11	imp 124	. . . . 5 ⊢ ((suc 𝑥 ∈ ω ∧ 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
13	10, 12	sylan 283	. . . 4 ⊢ ((𝑥 ∈ ω ∧ 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
14	13	rexlimiva 2617	. . 3 ⊢ (∃𝑥 ∈ ω 𝐴 = suc 𝑥 → 𝐴 ∈ ω)
15	9, 14	jaoi 717	. 2 ⊢ ((𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥) → 𝐴 ∈ ω)
16	6, 15	impbii 126	1 ⊢ (𝐴 ∈ ω ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ ω 𝐴 = suc 𝑥))

Syntax hints:   → wi 4   ↔ wb 105   ∨ wo 709   = wceq 1372   ∈ wcel 2175  ∃wrex 2484   ⊆ wss 3165  ∅c0 3459  suc csuc 4410  ωcom 4636

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-nul 4169  ax-pr 4252  ax-un 4478  ax-bd0 15613  ax-bdim 15614  ax-bdan 15615  ax-bdor 15616  ax-bdn 15617  ax-bdal 15618  ax-bdex 15619  ax-bdeq 15620  ax-bdel 15621  ax-bdsb 15622  ax-bdsep 15684  ax-infvn 15741

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-sn 3638  df-pr 3639  df-uni 3850  df-int 3885  df-suc 4416  df-iom 4637  df-bdc 15641  df-bj-ind 15727

This theorem is referenced by:  bj-findis  15779