Theorem bj-nn0suc 15694

Description: Proof of (biconditional form of) nn0suc 4641 from the core axioms of CZF. See also bj-nn0sucALT 15708. 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 15680	. . 3 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ 𝐴 𝐴 = suc 𝑥))
2		bj-omtrans 15686	. . . . 5 ⊢ (𝐴 ∈ ω → 𝐴 ⊆ ω)
3		ssrexv 3249	. . . . 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 4631	. . . 4 ⊢ ∅ ∈ ω
8		eleq1 2259	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 ∈ ω ↔ ∅ ∈ ω))
9	7, 8	mpbiri 168	. . 3 ⊢ (𝐴 = ∅ → 𝐴 ∈ ω)
10		bj-peano2 15669	. . . . 5 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
11		eleq1a 2268	. . . . . 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 2609	. . 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 1364   ∈ wcel 2167  ∃wrex 2476   ⊆ wss 3157  ∅c0 3451  suc csuc 4401  ωcom 4627

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-nul 4160  ax-pr 4243  ax-un 4469  ax-bd0 15543  ax-bdim 15544  ax-bdan 15545  ax-bdor 15546  ax-bdn 15547  ax-bdal 15548  ax-bdex 15549  ax-bdeq 15550  ax-bdel 15551  ax-bdsb 15552  ax-bdsep 15614  ax-infvn 15671

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-sn 3629  df-pr 3630  df-uni 3841  df-int 3876  df-suc 4407  df-iom 4628  df-bdc 15571  df-bj-ind 15657

This theorem is referenced by:  bj-findis  15709