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Theorem bj-indind 9929
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 9924 . . . 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 522 . . . 4 (((∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴) ∧ (∅ ∈ 𝐵 ∧ ∀𝑥𝐴 (𝑥𝐵 → suc 𝑥𝐵))) → ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) ∧ (∀𝑥𝐴 suc 𝑥𝐴 ∧ ∀𝑥𝐴 (𝑥𝐵 → suc 𝑥𝐵))))
41, 3sylanb 268 . . 3 ((Ind 𝐴 ∧ (∅ ∈ 𝐵 ∧ ∀𝑥𝐴 (𝑥𝐵 → suc 𝑥𝐵))) → ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) ∧ (∀𝑥𝐴 suc 𝑥𝐴 ∧ ∀𝑥𝐴 (𝑥𝐵 → suc 𝑥𝐵))))
5 elin 3123 . . . . 5 (∅ ∈ (𝐴𝐵) ↔ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵))
65biimpri 124 . . . 4 ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) → ∅ ∈ (𝐴𝐵))
7 r19.26 2438 . . . . . . . 8 (∀𝑥𝐴 (suc 𝑥𝐴 ∧ (𝑥𝐵 → suc 𝑥𝐵)) ↔ (∀𝑥𝐴 suc 𝑥𝐴 ∧ ∀𝑥𝐴 (𝑥𝐵 → suc 𝑥𝐵)))
87biimpri 124 . . . . . . 7 ((∀𝑥𝐴 suc 𝑥𝐴 ∧ ∀𝑥𝐴 (𝑥𝐵 → suc 𝑥𝐵)) → ∀𝑥𝐴 (suc 𝑥𝐴 ∧ (𝑥𝐵 → suc 𝑥𝐵)))
9 simpl 102 . . . . . . . . 9 ((suc 𝑥𝐴 ∧ (𝑥𝐵 → suc 𝑥𝐵)) → suc 𝑥𝐴)
10 simpr 103 . . . . . . . . 9 ((suc 𝑥𝐴 ∧ (𝑥𝐵 → suc 𝑥𝐵)) → (𝑥𝐵 → suc 𝑥𝐵))
11 elin 3123 . . . . . . . . . 10 (suc 𝑥 ∈ (𝐴𝐵) ↔ (suc 𝑥𝐴 ∧ suc 𝑥𝐵))
1211biimpri 124 . . . . . . . . 9 ((suc 𝑥𝐴 ∧ suc 𝑥𝐵) → suc 𝑥 ∈ (𝐴𝐵))
139, 10, 12syl6an 1323 . . . . . . . 8 ((suc 𝑥𝐴 ∧ (𝑥𝐵 → suc 𝑥𝐵)) → (𝑥𝐵 → suc 𝑥 ∈ (𝐴𝐵)))
1413ralimi 2381 . . . . . . 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 2308 . . . . . . 7 (∀𝑥𝐴 (𝑥𝐵 → suc 𝑥 ∈ (𝐴𝐵)) ↔ ∀𝑥(𝑥𝐴 → (𝑥𝐵 → suc 𝑥 ∈ (𝐴𝐵))))
17 elin 3123 . . . . . . . . 9 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
18 pm3.31 249 . . . . . . . . 9 ((𝑥𝐴 → (𝑥𝐵 → suc 𝑥 ∈ (𝐴𝐵))) → ((𝑥𝐴𝑥𝐵) → suc 𝑥 ∈ (𝐴𝐵)))
1917, 18syl5bi 141 . . . . . . . 8 ((𝑥𝐴 → (𝑥𝐵 → suc 𝑥 ∈ (𝐴𝐵))) → (𝑥 ∈ (𝐴𝐵) → suc 𝑥 ∈ (𝐴𝐵)))
2019alimi 1344 . . . . . . 7 (∀𝑥(𝑥𝐴 → (𝑥𝐵 → suc 𝑥 ∈ (𝐴𝐵))) → ∀𝑥(𝑥 ∈ (𝐴𝐵) → suc 𝑥 ∈ (𝐴𝐵)))
2116, 20sylbi 114 . . . . . 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 2308 . . . . 5 (∀𝑥 ∈ (𝐴𝐵)suc 𝑥 ∈ (𝐴𝐵) ↔ ∀𝑥(𝑥 ∈ (𝐴𝐵) → suc 𝑥 ∈ (𝐴𝐵)))
2422, 23sylibr 137 . . . 4 ((∀𝑥𝐴 suc 𝑥𝐴 ∧ ∀𝑥𝐴 (𝑥𝐵 → suc 𝑥𝐵)) → ∀𝑥 ∈ (𝐴𝐵)suc 𝑥 ∈ (𝐴𝐵))
256, 24anim12i 321 . . 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 9924 . 2 (Ind (𝐴𝐵) ↔ (∅ ∈ (𝐴𝐵) ∧ ∀𝑥 ∈ (𝐴𝐵)suc 𝑥 ∈ (𝐴𝐵)))
2826, 27sylibr 137 1 ((Ind 𝐴 ∧ (∅ ∈ 𝐵 ∧ ∀𝑥𝐴 (𝑥𝐵 → suc 𝑥𝐵))) → Ind (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wal 1241  wcel 1393  wral 2303  cin 2913  c0 3221  suc csuc 4098  Ind wind 9923
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-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-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
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 2308  df-v 2556  df-in 2921  df-bj-ind 9924
This theorem is referenced by:  peano5set  9937
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