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Theorem bj-indind 13119
Description: If 𝐴 is inductive and 𝐵 is "inductive in 𝐴", then (𝐴𝐵) is inductive. (Contributed by BJ, 25-Oct-2020.)
Assertion
Ref Expression
bj-indind ((Ind 𝐴 ∧ (∅ ∈ 𝐵 ∧ ∀𝑥𝐴 (𝑥𝐵 → suc 𝑥𝐵))) → Ind (𝐴𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem bj-indind
StepHypRef Expression
1 df-bj-ind 13114 . . . 4 (Ind 𝐴 ↔ (∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴))
2 id 19 . . . . 5 (((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) ∧ (∀𝑥𝐴 suc 𝑥𝐴 ∧ ∀𝑥𝐴 (𝑥𝐵 → suc 𝑥𝐵))) → ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) ∧ (∀𝑥𝐴 suc 𝑥𝐴 ∧ ∀𝑥𝐴 (𝑥𝐵 → suc 𝑥𝐵))))
32an4s 577 . . . 4 (((∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴) ∧ (∅ ∈ 𝐵 ∧ ∀𝑥𝐴 (𝑥𝐵 → suc 𝑥𝐵))) → ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) ∧ (∀𝑥𝐴 suc 𝑥𝐴 ∧ ∀𝑥𝐴 (𝑥𝐵 → suc 𝑥𝐵))))
41, 3sylanb 282 . . 3 ((Ind 𝐴 ∧ (∅ ∈ 𝐵 ∧ ∀𝑥𝐴 (𝑥𝐵 → suc 𝑥𝐵))) → ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) ∧ (∀𝑥𝐴 suc 𝑥𝐴 ∧ ∀𝑥𝐴 (𝑥𝐵 → suc 𝑥𝐵))))
5 elin 3254 . . . . 5 (∅ ∈ (𝐴𝐵) ↔ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵))
65biimpri 132 . . . 4 ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) → ∅ ∈ (𝐴𝐵))
7 r19.26 2556 . . . . . . . 8 (∀𝑥𝐴 (suc 𝑥𝐴 ∧ (𝑥𝐵 → suc 𝑥𝐵)) ↔ (∀𝑥𝐴 suc 𝑥𝐴 ∧ ∀𝑥𝐴 (𝑥𝐵 → suc 𝑥𝐵)))
87biimpri 132 . . . . . . 7 ((∀𝑥𝐴 suc 𝑥𝐴 ∧ ∀𝑥𝐴 (𝑥𝐵 → suc 𝑥𝐵)) → ∀𝑥𝐴 (suc 𝑥𝐴 ∧ (𝑥𝐵 → suc 𝑥𝐵)))
9 simpl 108 . . . . . . . . 9 ((suc 𝑥𝐴 ∧ (𝑥𝐵 → suc 𝑥𝐵)) → suc 𝑥𝐴)
10 simpr 109 . . . . . . . . 9 ((suc 𝑥𝐴 ∧ (𝑥𝐵 → suc 𝑥𝐵)) → (𝑥𝐵 → suc 𝑥𝐵))
11 elin 3254 . . . . . . . . . 10 (suc 𝑥 ∈ (𝐴𝐵) ↔ (suc 𝑥𝐴 ∧ suc 𝑥𝐵))
1211biimpri 132 . . . . . . . . 9 ((suc 𝑥𝐴 ∧ suc 𝑥𝐵) → suc 𝑥 ∈ (𝐴𝐵))
139, 10, 12syl6an 1410 . . . . . . . 8 ((suc 𝑥𝐴 ∧ (𝑥𝐵 → suc 𝑥𝐵)) → (𝑥𝐵 → suc 𝑥 ∈ (𝐴𝐵)))
1413ralimi 2493 . . . . . . 7 (∀𝑥𝐴 (suc 𝑥𝐴 ∧ (𝑥𝐵 → suc 𝑥𝐵)) → ∀𝑥𝐴 (𝑥𝐵 → suc 𝑥 ∈ (𝐴𝐵)))
158, 14syl 14 . . . . . 6 ((∀𝑥𝐴 suc 𝑥𝐴 ∧ ∀𝑥𝐴 (𝑥𝐵 → suc 𝑥𝐵)) → ∀𝑥𝐴 (𝑥𝐵 → suc 𝑥 ∈ (𝐴𝐵)))
16 df-ral 2419 . . . . . . 7 (∀𝑥𝐴 (𝑥𝐵 → suc 𝑥 ∈ (𝐴𝐵)) ↔ ∀𝑥(𝑥𝐴 → (𝑥𝐵 → suc 𝑥 ∈ (𝐴𝐵))))
17 elin 3254 . . . . . . . . 9 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
18 pm3.31 260 . . . . . . . . 9 ((𝑥𝐴 → (𝑥𝐵 → suc 𝑥 ∈ (𝐴𝐵))) → ((𝑥𝐴𝑥𝐵) → suc 𝑥 ∈ (𝐴𝐵)))
1917, 18syl5bi 151 . . . . . . . 8 ((𝑥𝐴 → (𝑥𝐵 → suc 𝑥 ∈ (𝐴𝐵))) → (𝑥 ∈ (𝐴𝐵) → suc 𝑥 ∈ (𝐴𝐵)))
2019alimi 1431 . . . . . . 7 (∀𝑥(𝑥𝐴 → (𝑥𝐵 → suc 𝑥 ∈ (𝐴𝐵))) → ∀𝑥(𝑥 ∈ (𝐴𝐵) → suc 𝑥 ∈ (𝐴𝐵)))
2116, 20sylbi 120 . . . . . 6 (∀𝑥𝐴 (𝑥𝐵 → suc 𝑥 ∈ (𝐴𝐵)) → ∀𝑥(𝑥 ∈ (𝐴𝐵) → suc 𝑥 ∈ (𝐴𝐵)))
2215, 21syl 14 . . . . 5 ((∀𝑥𝐴 suc 𝑥𝐴 ∧ ∀𝑥𝐴 (𝑥𝐵 → suc 𝑥𝐵)) → ∀𝑥(𝑥 ∈ (𝐴𝐵) → suc 𝑥 ∈ (𝐴𝐵)))
23 df-ral 2419 . . . . 5 (∀𝑥 ∈ (𝐴𝐵)suc 𝑥 ∈ (𝐴𝐵) ↔ ∀𝑥(𝑥 ∈ (𝐴𝐵) → suc 𝑥 ∈ (𝐴𝐵)))
2422, 23sylibr 133 . . . 4 ((∀𝑥𝐴 suc 𝑥𝐴 ∧ ∀𝑥𝐴 (𝑥𝐵 → suc 𝑥𝐵)) → ∀𝑥 ∈ (𝐴𝐵)suc 𝑥 ∈ (𝐴𝐵))
256, 24anim12i 336 . . 3 (((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) ∧ (∀𝑥𝐴 suc 𝑥𝐴 ∧ ∀𝑥𝐴 (𝑥𝐵 → suc 𝑥𝐵))) → (∅ ∈ (𝐴𝐵) ∧ ∀𝑥 ∈ (𝐴𝐵)suc 𝑥 ∈ (𝐴𝐵)))
264, 25syl 14 . 2 ((Ind 𝐴 ∧ (∅ ∈ 𝐵 ∧ ∀𝑥𝐴 (𝑥𝐵 → suc 𝑥𝐵))) → (∅ ∈ (𝐴𝐵) ∧ ∀𝑥 ∈ (𝐴𝐵)suc 𝑥 ∈ (𝐴𝐵)))
27 df-bj-ind 13114 . 2 (Ind (𝐴𝐵) ↔ (∅ ∈ (𝐴𝐵) ∧ ∀𝑥 ∈ (𝐴𝐵)suc 𝑥 ∈ (𝐴𝐵)))
2826, 27sylibr 133 1 ((Ind 𝐴 ∧ (∅ ∈ 𝐵 ∧ ∀𝑥𝐴 (𝑥𝐵 → suc 𝑥𝐵))) → Ind (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1329  wcel 1480  wral 2414  cin 3065  c0 3358  suc csuc 4282  Ind wind 13113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-v 2683  df-in 3072  df-bj-ind 13114
This theorem is referenced by:  peano5set  13127
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