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Theorem peano5setOLD 10065
 Description: Obsolete version of peano5set 10064 as of 26-Oct-2020. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
peano5setOLD ((ω ∩ 𝐴) ∈ 𝑉 → ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) → ω ⊆ 𝐴))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem peano5setOLD
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 bj-dfom 10057 . . . 4 ω = {𝑦 ∣ Ind 𝑦}
2 peano1 4317 . . . . . . . . . . 11 ∅ ∈ ω
3 elin 3126 . . . . . . . . . . 11 (∅ ∈ (ω ∩ 𝐴) ↔ (∅ ∈ ω ∧ ∅ ∈ 𝐴))
42, 3mpbiran 847 . . . . . . . . . 10 (∅ ∈ (ω ∩ 𝐴) ↔ ∅ ∈ 𝐴)
54biimpri 124 . . . . . . . . 9 (∅ ∈ 𝐴 → ∅ ∈ (ω ∩ 𝐴))
6 bj-peano2 10063 . . . . . . . . . . . . . . 15 (𝑥 ∈ ω → suc 𝑥 ∈ ω)
76adantr 261 . . . . . . . . . . . . . 14 ((𝑥 ∈ ω ∧ 𝑥𝐴) → suc 𝑥 ∈ ω)
87a1i 9 . . . . . . . . . . . . 13 ((𝑥 ∈ ω → (𝑥𝐴 → suc 𝑥𝐴)) → ((𝑥 ∈ ω ∧ 𝑥𝐴) → suc 𝑥 ∈ ω))
9 pm3.31 249 . . . . . . . . . . . . 13 ((𝑥 ∈ ω → (𝑥𝐴 → suc 𝑥𝐴)) → ((𝑥 ∈ ω ∧ 𝑥𝐴) → suc 𝑥𝐴))
108, 9jcad 291 . . . . . . . . . . . 12 ((𝑥 ∈ ω → (𝑥𝐴 → suc 𝑥𝐴)) → ((𝑥 ∈ ω ∧ 𝑥𝐴) → (suc 𝑥 ∈ ω ∧ suc 𝑥𝐴)))
1110alimi 1344 . . . . . . . . . . 11 (∀𝑥(𝑥 ∈ ω → (𝑥𝐴 → suc 𝑥𝐴)) → ∀𝑥((𝑥 ∈ ω ∧ 𝑥𝐴) → (suc 𝑥 ∈ ω ∧ suc 𝑥𝐴)))
12 df-ral 2311 . . . . . . . . . . 11 (∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴) ↔ ∀𝑥(𝑥 ∈ ω → (𝑥𝐴 → suc 𝑥𝐴)))
13 elin 3126 . . . . . . . . . . . . 13 (𝑥 ∈ (ω ∩ 𝐴) ↔ (𝑥 ∈ ω ∧ 𝑥𝐴))
14 elin 3126 . . . . . . . . . . . . 13 (suc 𝑥 ∈ (ω ∩ 𝐴) ↔ (suc 𝑥 ∈ ω ∧ suc 𝑥𝐴))
1513, 14imbi12i 228 . . . . . . . . . . . 12 ((𝑥 ∈ (ω ∩ 𝐴) → suc 𝑥 ∈ (ω ∩ 𝐴)) ↔ ((𝑥 ∈ ω ∧ 𝑥𝐴) → (suc 𝑥 ∈ ω ∧ suc 𝑥𝐴)))
1615albii 1359 . . . . . . . . . . 11 (∀𝑥(𝑥 ∈ (ω ∩ 𝐴) → suc 𝑥 ∈ (ω ∩ 𝐴)) ↔ ∀𝑥((𝑥 ∈ ω ∧ 𝑥𝐴) → (suc 𝑥 ∈ ω ∧ suc 𝑥𝐴)))
1711, 12, 163imtr4i 190 . . . . . . . . . 10 (∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴) → ∀𝑥(𝑥 ∈ (ω ∩ 𝐴) → suc 𝑥 ∈ (ω ∩ 𝐴)))
18 df-ral 2311 . . . . . . . . . 10 (∀𝑥 ∈ (ω ∩ 𝐴)suc 𝑥 ∈ (ω ∩ 𝐴) ↔ ∀𝑥(𝑥 ∈ (ω ∩ 𝐴) → suc 𝑥 ∈ (ω ∩ 𝐴)))
1917, 18sylibr 137 . . . . . . . . 9 (∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴) → ∀𝑥 ∈ (ω ∩ 𝐴)suc 𝑥 ∈ (ω ∩ 𝐴))
205, 19anim12i 321 . . . . . . . 8 ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) → (∅ ∈ (ω ∩ 𝐴) ∧ ∀𝑥 ∈ (ω ∩ 𝐴)suc 𝑥 ∈ (ω ∩ 𝐴)))
21 df-bj-ind 10051 . . . . . . . 8 (Ind (ω ∩ 𝐴) ↔ (∅ ∈ (ω ∩ 𝐴) ∧ ∀𝑥 ∈ (ω ∩ 𝐴)suc 𝑥 ∈ (ω ∩ 𝐴)))
2220, 21sylibr 137 . . . . . . 7 ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) → Ind (ω ∩ 𝐴))
23 bj-indeq 10053 . . . . . . . 8 (𝑦 = (ω ∩ 𝐴) → (Ind 𝑦 ↔ Ind (ω ∩ 𝐴)))
2423elabg 2688 . . . . . . 7 ((ω ∩ 𝐴) ∈ 𝑉 → ((ω ∩ 𝐴) ∈ {𝑦 ∣ Ind 𝑦} ↔ Ind (ω ∩ 𝐴)))
2522, 24syl5ibr 145 . . . . . 6 ((ω ∩ 𝐴) ∈ 𝑉 → ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) → (ω ∩ 𝐴) ∈ {𝑦 ∣ Ind 𝑦}))
2625imp 115 . . . . 5 (((ω ∩ 𝐴) ∈ 𝑉 ∧ (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴))) → (ω ∩ 𝐴) ∈ {𝑦 ∣ Ind 𝑦})
27 intss1 3630 . . . . 5 ((ω ∩ 𝐴) ∈ {𝑦 ∣ Ind 𝑦} → {𝑦 ∣ Ind 𝑦} ⊆ (ω ∩ 𝐴))
2826, 27syl 14 . . . 4 (((ω ∩ 𝐴) ∈ 𝑉 ∧ (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴))) → {𝑦 ∣ Ind 𝑦} ⊆ (ω ∩ 𝐴))
291, 28syl5eqss 2989 . . 3 (((ω ∩ 𝐴) ∈ 𝑉 ∧ (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴))) → ω ⊆ (ω ∩ 𝐴))
30 inss2 3158 . . 3 (ω ∩ 𝐴) ⊆ 𝐴
3129, 30syl6ss 2957 . 2 (((ω ∩ 𝐴) ∈ 𝑉 ∧ (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴))) → ω ⊆ 𝐴)
3231ex 108 1 ((ω ∩ 𝐴) ∈ 𝑉 → ((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ ω (𝑥𝐴 → suc 𝑥𝐴)) → ω ⊆ 𝐴))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1241   ∈ wcel 1393  {cab 2026  ∀wral 2306   ∩ cin 2916   ⊆ wss 2917  ∅c0 3224  ∩ cint 3615  suc csuc 4102  ωcom 4313  Ind wind 10050 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 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-nul 3883  ax-pr 3944  ax-un 4170  ax-bd0 9933  ax-bdor 9936  ax-bdex 9939  ax-bdeq 9940  ax-bdel 9941  ax-bdsb 9942  ax-bdsep 10004 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-sn 3381  df-pr 3382  df-uni 3581  df-int 3616  df-suc 4108  df-iom 4314  df-bdc 9961  df-bj-ind 10051 This theorem is referenced by: (None)
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