Theorem bj-nn0suc 16904

Description: Proof of (biconditional form of) nn0suc 4746 from the core axioms of CZF. See also bj-nn0sucALT 16918. 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 16890	. . 3 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∃𝑥 ∈ 𝐴 𝐴 = suc 𝑥))
2		bj-omtrans 16896	. . . . 5 ⊢ (𝐴 ∈ ω → 𝐴 ⊆ ω)
3		ssrexv 3313	. . . . 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 800	. . 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 4736	. . . 4 ⊢ ∅ ∈ ω
8		eleq1 2301	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 ∈ ω ↔ ∅ ∈ ω))
9	7, 8	mpbiri 168	. . 3 ⊢ (𝐴 = ∅ → 𝐴 ∈ ω)
10		bj-peano2 16879	. . . . 5 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
11		eleq1a 2310	. . . . . 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 2663	. . 3 ⊢ (∃𝑥 ∈ ω 𝐴 = suc 𝑥 → 𝐴 ∈ ω)
15	9, 14	jaoi 728	. 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 720   = wceq 1402   ∈ wcel 2209  ∃wrex 2529   ⊆ wss 3220  ∅c0 3520  suc csuc 4505  ωcom 4732

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-nul 4254  ax-pr 4341  ax-un 4573  ax-bd0 16753  ax-bdim 16754  ax-bdan 16755  ax-bdor 16756  ax-bdn 16757  ax-bdal 16758  ax-bdex 16759  ax-bdeq 16760  ax-bdel 16761  ax-bdsb 16762  ax-bdsep 16824  ax-infvn 16881

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-sn 3711  df-pr 3712  df-uni 3931  df-int 3966  df-suc 4511  df-iom 4733  df-bdc 16781  df-bj-ind 16867

This theorem is referenced by:  bj-findis  16919