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Theorem bj-indind 14687
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 14682 . . . 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 588 . . . 4 (((∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴) ∧ (∅ ∈ 𝐵 ∧ ∀𝑥𝐴 (𝑥𝐵 → suc 𝑥𝐵))) → ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) ∧ (∀𝑥𝐴 suc 𝑥𝐴 ∧ ∀𝑥𝐴 (𝑥𝐵 → suc 𝑥𝐵))))
41, 3sylanb 284 . . 3 ((Ind 𝐴 ∧ (∅ ∈ 𝐵 ∧ ∀𝑥𝐴 (𝑥𝐵 → suc 𝑥𝐵))) → ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) ∧ (∀𝑥𝐴 suc 𝑥𝐴 ∧ ∀𝑥𝐴 (𝑥𝐵 → suc 𝑥𝐵))))
5 elin 3319 . . . . 5 (∅ ∈ (𝐴𝐵) ↔ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵))
65biimpri 133 . . . 4 ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) → ∅ ∈ (𝐴𝐵))
7 r19.26 2603 . . . . . . . 8 (∀𝑥𝐴 (suc 𝑥𝐴 ∧ (𝑥𝐵 → suc 𝑥𝐵)) ↔ (∀𝑥𝐴 suc 𝑥𝐴 ∧ ∀𝑥𝐴 (𝑥𝐵 → suc 𝑥𝐵)))
87biimpri 133 . . . . . . 7 ((∀𝑥𝐴 suc 𝑥𝐴 ∧ ∀𝑥𝐴 (𝑥𝐵 → suc 𝑥𝐵)) → ∀𝑥𝐴 (suc 𝑥𝐴 ∧ (𝑥𝐵 → suc 𝑥𝐵)))
9 simpl 109 . . . . . . . . 9 ((suc 𝑥𝐴 ∧ (𝑥𝐵 → suc 𝑥𝐵)) → suc 𝑥𝐴)
10 simpr 110 . . . . . . . . 9 ((suc 𝑥𝐴 ∧ (𝑥𝐵 → suc 𝑥𝐵)) → (𝑥𝐵 → suc 𝑥𝐵))
11 elin 3319 . . . . . . . . . 10 (suc 𝑥 ∈ (𝐴𝐵) ↔ (suc 𝑥𝐴 ∧ suc 𝑥𝐵))
1211biimpri 133 . . . . . . . . 9 ((suc 𝑥𝐴 ∧ suc 𝑥𝐵) → suc 𝑥 ∈ (𝐴𝐵))
139, 10, 12syl6an 1434 . . . . . . . 8 ((suc 𝑥𝐴 ∧ (𝑥𝐵 → suc 𝑥𝐵)) → (𝑥𝐵 → suc 𝑥 ∈ (𝐴𝐵)))
1413ralimi 2540 . . . . . . 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 2460 . . . . . . 7 (∀𝑥𝐴 (𝑥𝐵 → suc 𝑥 ∈ (𝐴𝐵)) ↔ ∀𝑥(𝑥𝐴 → (𝑥𝐵 → suc 𝑥 ∈ (𝐴𝐵))))
17 elin 3319 . . . . . . . . 9 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
18 pm3.31 262 . . . . . . . . 9 ((𝑥𝐴 → (𝑥𝐵 → suc 𝑥 ∈ (𝐴𝐵))) → ((𝑥𝐴𝑥𝐵) → suc 𝑥 ∈ (𝐴𝐵)))
1917, 18biimtrid 152 . . . . . . . 8 ((𝑥𝐴 → (𝑥𝐵 → suc 𝑥 ∈ (𝐴𝐵))) → (𝑥 ∈ (𝐴𝐵) → suc 𝑥 ∈ (𝐴𝐵)))
2019alimi 1455 . . . . . . 7 (∀𝑥(𝑥𝐴 → (𝑥𝐵 → suc 𝑥 ∈ (𝐴𝐵))) → ∀𝑥(𝑥 ∈ (𝐴𝐵) → suc 𝑥 ∈ (𝐴𝐵)))
2116, 20sylbi 121 . . . . . 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 2460 . . . . 5 (∀𝑥 ∈ (𝐴𝐵)suc 𝑥 ∈ (𝐴𝐵) ↔ ∀𝑥(𝑥 ∈ (𝐴𝐵) → suc 𝑥 ∈ (𝐴𝐵)))
2422, 23sylibr 134 . . . 4 ((∀𝑥𝐴 suc 𝑥𝐴 ∧ ∀𝑥𝐴 (𝑥𝐵 → suc 𝑥𝐵)) → ∀𝑥 ∈ (𝐴𝐵)suc 𝑥 ∈ (𝐴𝐵))
256, 24anim12i 338 . . 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 14682 . 2 (Ind (𝐴𝐵) ↔ (∅ ∈ (𝐴𝐵) ∧ ∀𝑥 ∈ (𝐴𝐵)suc 𝑥 ∈ (𝐴𝐵)))
2826, 27sylibr 134 1 ((Ind 𝐴 ∧ (∅ ∈ 𝐵 ∧ ∀𝑥𝐴 (𝑥𝐵 → suc 𝑥𝐵))) → Ind (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wal 1351  wcel 2148  wral 2455  cin 3129  c0 3423  suc csuc 4366  Ind wind 14681
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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2740  df-in 3136  df-bj-ind 14682
This theorem is referenced by:  peano5set  14695
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