Theorem bj-nn0suc 16495

Description: Proof of (biconditional form of) nn0suc 4700 from the core axioms of CZF. See also bj-nn0sucALT 16509. 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 16481	. . 3 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ 𝐴 𝐴 = suc 𝑥))
2		bj-omtrans 16487	. . . . 5 ⊢ (𝐴 ∈ ω → 𝐴 ⊆ ω)
3		ssrexv 3290	. . . . 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 793	. . 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 4690	. . . 4 ⊢ ∅ ∈ ω
8		eleq1 2292	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 ∈ ω ↔ ∅ ∈ ω))
9	7, 8	mpbiri 168	. . 3 ⊢ (𝐴 = ∅ → 𝐴 ∈ ω)
10		bj-peano2 16470	. . . . 5 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
11		eleq1a 2301	. . . . . 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 2643	. . 3 ⊢ (∃𝑥 ∈ ω 𝐴 = suc 𝑥 → 𝐴 ∈ ω)
15	9, 14	jaoi 721	. 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 713   = wceq 1395   ∈ wcel 2200  ∃wrex 2509   ⊆ wss 3198  ∅c0 3492  suc csuc 4460  ωcom 4686

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-nul 4213  ax-pr 4297  ax-un 4528  ax-bd0 16344  ax-bdim 16345  ax-bdan 16346  ax-bdor 16347  ax-bdn 16348  ax-bdal 16349  ax-bdex 16350  ax-bdeq 16351  ax-bdel 16352  ax-bdsb 16353  ax-bdsep 16415  ax-infvn 16472

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-sn 3673  df-pr 3674  df-uni 3892  df-int 3927  df-suc 4466  df-iom 4687  df-bdc 16372  df-bj-ind 16458

This theorem is referenced by:  bj-findis  16510