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Theorem peano5set 7162
Description: Version of peano5 4248 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
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 dfom3 4242 . . . 4 𝜔 = {y ∣ (∅ y x y suc x y)}
2 peano1 4244 . . . . . . . . . 10 𝜔
3 elin 3103 . . . . . . . . . 10 (∅ (𝜔 ∩ A) ↔ (∅ 𝜔 A))
42, 3mpbiran 835 . . . . . . . . 9 (∅ (𝜔 ∩ A) ↔ ∅ A)
54biimpri 124 . . . . . . . 8 (∅ A → ∅ (𝜔 ∩ A))
6 bj-peano2 7161 . . . . . . . . . . . . . 14 (x 𝜔 → suc x 𝜔)
76adantr 261 . . . . . . . . . . . . 13 ((x 𝜔 x A) → suc x 𝜔)
87a1i 9 . . . . . . . . . . . 12 ((x 𝜔 → (x A → suc x A)) → ((x 𝜔 x A) → suc x 𝜔))
9 pm3.31 249 . . . . . . . . . . . 12 ((x 𝜔 → (x A → suc x A)) → ((x 𝜔 x A) → suc x A))
108, 9jcad 291 . . . . . . . . . . 11 ((x 𝜔 → (x A → suc x A)) → ((x 𝜔 x A) → (suc x 𝜔 suc x A)))
1110alimi 1324 . . . . . . . . . 10 (x(x 𝜔 → (x A → suc x A)) → x((x 𝜔 x A) → (suc x 𝜔 suc x A)))
12 df-ral 2289 . . . . . . . . . 10 (x 𝜔 (x A → suc x A) ↔ x(x 𝜔 → (x A → suc x A)))
13 elin 3103 . . . . . . . . . . . 12 (x (𝜔 ∩ A) ↔ (x 𝜔 x A))
14 elin 3103 . . . . . . . . . . . 12 (suc x (𝜔 ∩ A) ↔ (suc x 𝜔 suc x A))
1513, 14imbi12i 228 . . . . . . . . . . 11 ((x (𝜔 ∩ A) → suc x (𝜔 ∩ A)) ↔ ((x 𝜔 x A) → (suc x 𝜔 suc x A)))
1615albii 1339 . . . . . . . . . 10 (x(x (𝜔 ∩ A) → suc x (𝜔 ∩ A)) ↔ x((x 𝜔 x A) → (suc x 𝜔 suc x A)))
1711, 12, 163imtr4i 190 . . . . . . . . 9 (x 𝜔 (x A → suc x A) → x(x (𝜔 ∩ A) → suc x (𝜔 ∩ A)))
18 df-ral 2289 . . . . . . . . 9 (x (𝜔 ∩ A)suc x (𝜔 ∩ A) ↔ x(x (𝜔 ∩ A) → suc x (𝜔 ∩ A)))
1917, 18sylibr 137 . . . . . . . 8 (x 𝜔 (x A → suc x A) → x (𝜔 ∩ A)suc x (𝜔 ∩ A))
205, 19anim12i 321 . . . . . . 7 ((∅ A x 𝜔 (x A → suc x A)) → (∅ (𝜔 ∩ A) x (𝜔 ∩ A)suc x (𝜔 ∩ A)))
21 eleq2 2083 . . . . . . . . 9 (y = (𝜔 ∩ A) → (∅ y ↔ ∅ (𝜔 ∩ A)))
22 eleq2 2083 . . . . . . . . . 10 (y = (𝜔 ∩ A) → (suc x y ↔ suc x (𝜔 ∩ A)))
2322raleqbi1dv 2491 . . . . . . . . 9 (y = (𝜔 ∩ A) → (x y suc x yx (𝜔 ∩ A)suc x (𝜔 ∩ A)))
2421, 23anbi12d 445 . . . . . . . 8 (y = (𝜔 ∩ A) → ((∅ y x y suc x y) ↔ (∅ (𝜔 ∩ A) x (𝜔 ∩ A)suc x (𝜔 ∩ A))))
2524elabg 2665 . . . . . . 7 ((𝜔 ∩ A) 𝑉 → ((𝜔 ∩ A) {y ∣ (∅ y x y suc x y)} ↔ (∅ (𝜔 ∩ A) x (𝜔 ∩ A)suc x (𝜔 ∩ A))))
2620, 25syl5ibr 145 . . . . . 6 ((𝜔 ∩ A) 𝑉 → ((∅ A x 𝜔 (x A → suc x A)) → (𝜔 ∩ A) {y ∣ (∅ y x y suc x y)}))
2726imp 115 . . . . 5 (((𝜔 ∩ A) 𝑉 (∅ A x 𝜔 (x A → suc x A))) → (𝜔 ∩ A) {y ∣ (∅ y x y suc x y)})
28 intss1 3604 . . . . 5 ((𝜔 ∩ A) {y ∣ (∅ y x y suc x y)} → {y ∣ (∅ y x y suc x y)} ⊆ (𝜔 ∩ A))
2927, 28syl 14 . . . 4 (((𝜔 ∩ A) 𝑉 (∅ A x 𝜔 (x A → suc x A))) → {y ∣ (∅ y x y suc x y)} ⊆ (𝜔 ∩ A))
301, 29syl5eqss 2966 . . 3 (((𝜔 ∩ A) 𝑉 (∅ A x 𝜔 (x A → suc x A))) → 𝜔 ⊆ (𝜔 ∩ A))
31 inss2 3135 . . 3 (𝜔 ∩ A) ⊆ A
3230, 31syl6ss 2934 . 2 (((𝜔 ∩ A) 𝑉 (∅ A x 𝜔 (x A → suc x A))) → 𝜔 ⊆ A)
3332ex 108 1 ((𝜔 ∩ A) 𝑉 → ((∅ A x 𝜔 (x A → suc x A)) → 𝜔 ⊆ A))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wal 1226   = wceq 1228   wcel 1374  {cab 2008  wral 2284  cin 2893  wss 2894  c0 3201   cint 3589  suc csuc 4051  𝜔com 4240
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-nul 3857  ax-pr 3918  ax-un 4120  ax-bd0 7040  ax-bdor 7043  ax-bdex 7046  ax-bdeq 7047  ax-bdel 7048  ax-bdsb 7049  ax-bdsep 7111
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-rab 2293  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-sn 3356  df-pr 3357  df-uni 3555  df-int 3590  df-suc 4057  df-iom 4241  df-bdc 7068  df-bj-ind 7150
This theorem is referenced by:  bdpeano5  7165  speano5  7166
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