Theorem peano5set 16233

Description: Version of peano5 4687 when ω ∩ 𝐴 is assumed to be a set, allowing a proof from the core axioms of CZF. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
peano5set	⊢ ((ω ∩ 𝐴) ∈ 𝑉 → ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴))

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem peano5set

Step	Hyp	Ref	Expression
1		bj-omind 16227	. . . . 5 ⊢ Ind ω
2		bj-indind 16225	. . . . 5 ⊢ ((Ind ω ∧ (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴))) → Ind (ω ∩ 𝐴))
3	1, 2	mpan 424	. . . 4 ⊢ ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → Ind (ω ∩ 𝐴))
4		bj-omssind 16228	. . . . 5 ⊢ ((ω ∩ 𝐴) ∈ 𝑉 → (Ind (ω ∩ 𝐴) → ω ⊆ (ω ∩ 𝐴)))
5	4	imp 124	. . . 4 ⊢ (((ω ∩ 𝐴) ∈ 𝑉 ∧ Ind (ω ∩ 𝐴)) → ω ⊆ (ω ∩ 𝐴))
6	3, 5	sylan2 286	. . 3 ⊢ (((ω ∩ 𝐴) ∈ 𝑉 ∧ (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴))) → ω ⊆ (ω ∩ 𝐴))
7		inss2 3425	. . 3 ⊢ (ω ∩ 𝐴) ⊆ 𝐴
8	6, 7	sstrdi 3236	. 2 ⊢ (((ω ∩ 𝐴) ∈ 𝑉 ∧ (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴))) → ω ⊆ 𝐴)
9	8	ex 115	1 ⊢ ((ω ∩ 𝐴) ∈ 𝑉 → ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥 ∈ 𝐴 → suc 𝑥 ∈ 𝐴)) → ω ⊆ 𝐴))

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2200  ∀wral 2508   ∩ cin 3196   ⊆ wss 3197  ∅c0 3491  suc csuc 4453  ωcom 4679  Ind wind 16219

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 4209  ax-pr 4292  ax-un 4521  ax-bd0 16106  ax-bdor 16109  ax-bdex 16112  ax-bdeq 16113  ax-bdel 16114  ax-bdsb 16115  ax-bdsep 16177

This theorem depends on definitions:  df-bi 117  df-tru 1398  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 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-sn 3672  df-pr 3673  df-uni 3888  df-int 3923  df-suc 4459  df-iom 4680  df-bdc 16134  df-bj-ind 16220

This theorem is referenced by:  bdpeano5  16236  speano5  16237