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Theorem bj-indind 9391
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 x A (x B → suc x B))) → Ind (AB))
Distinct variable groups:   x,A   x,B

Proof of Theorem bj-indind
StepHypRef Expression
1 df-bj-ind 9386 . . . 4 (Ind A ↔ (∅ A x A suc x A))
2 id 19 . . . . 5 (((∅ A B) (x A suc x A x A (x B → suc x B))) → ((∅ A B) (x A suc x A x A (x B → suc x B))))
32an4s 522 . . . 4 (((∅ A x A suc x A) (∅ B x A (x B → suc x B))) → ((∅ A B) (x A suc x A x A (x B → suc x B))))
41, 3sylanb 268 . . 3 ((Ind A (∅ B x A (x B → suc x B))) → ((∅ A B) (x A suc x A x A (x B → suc x B))))
5 elin 3120 . . . . 5 (∅ (AB) ↔ (∅ A B))
65biimpri 124 . . . 4 ((∅ A B) → ∅ (AB))
7 r19.26 2435 . . . . . . . 8 (x A (suc x A (x B → suc x B)) ↔ (x A suc x A x A (x B → suc x B)))
87biimpri 124 . . . . . . 7 ((x A suc x A x A (x B → suc x B)) → x A (suc x A (x B → suc x B)))
9 simpl 102 . . . . . . . . 9 ((suc x A (x B → suc x B)) → suc x A)
10 simpr 103 . . . . . . . . 9 ((suc x A (x B → suc x B)) → (x B → suc x B))
11 elin 3120 . . . . . . . . . 10 (suc x (AB) ↔ (suc x A suc x B))
1211biimpri 124 . . . . . . . . 9 ((suc x A suc x B) → suc x (AB))
139, 10, 12syl6an 1320 . . . . . . . 8 ((suc x A (x B → suc x B)) → (x B → suc x (AB)))
1413ralimi 2378 . . . . . . 7 (x A (suc x A (x B → suc x B)) → x A (x B → suc x (AB)))
158, 14syl 14 . . . . . 6 ((x A suc x A x A (x B → suc x B)) → x A (x B → suc x (AB)))
16 df-ral 2305 . . . . . . 7 (x A (x B → suc x (AB)) ↔ x(x A → (x B → suc x (AB))))
17 elin 3120 . . . . . . . . 9 (x (AB) ↔ (x A x B))
18 pm3.31 249 . . . . . . . . 9 ((x A → (x B → suc x (AB))) → ((x A x B) → suc x (AB)))
1917, 18syl5bi 141 . . . . . . . 8 ((x A → (x B → suc x (AB))) → (x (AB) → suc x (AB)))
2019alimi 1341 . . . . . . 7 (x(x A → (x B → suc x (AB))) → x(x (AB) → suc x (AB)))
2116, 20sylbi 114 . . . . . 6 (x A (x B → suc x (AB)) → x(x (AB) → suc x (AB)))
2215, 21syl 14 . . . . 5 ((x A suc x A x A (x B → suc x B)) → x(x (AB) → suc x (AB)))
23 df-ral 2305 . . . . 5 (x (AB)suc x (AB) ↔ x(x (AB) → suc x (AB)))
2422, 23sylibr 137 . . . 4 ((x A suc x A x A (x B → suc x B)) → x (AB)suc x (AB))
256, 24anim12i 321 . . 3 (((∅ A B) (x A suc x A x A (x B → suc x B))) → (∅ (AB) x (AB)suc x (AB)))
264, 25syl 14 . 2 ((Ind A (∅ B x A (x B → suc x B))) → (∅ (AB) x (AB)suc x (AB)))
27 df-bj-ind 9386 . 2 (Ind (AB) ↔ (∅ (AB) x (AB)suc x (AB)))
2826, 27sylibr 137 1 ((Ind A (∅ B x A (x B → suc x B))) → Ind (AB))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wal 1240   wcel 1390  wral 2300  cin 2910  c0 3218  suc csuc 4068  Ind wind 9385
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-in 2918  df-bj-ind 9386
This theorem is referenced by:  peano5set  9399
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