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Theorem bj-indint 9320
Description: The property of being an inductive class is closed under intersections. (Contributed by BJ, 30-Nov-2019.)
Assertion
Ref Expression
bj-indint Ind {x A ∣ Ind x}
Distinct variable group:   x,A

Proof of Theorem bj-indint
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 df-bj-ind 9316 . . . . 5 (Ind x ↔ (∅ x y x suc y x))
21simplbi 259 . . . 4 (Ind x → ∅ x)
32rgenw 2370 . . 3 x A (Ind x → ∅ x)
4 0ex 3875 . . . 4 V
54elintrab 3618 . . 3 (∅ {x A ∣ Ind x} ↔ x A (Ind x → ∅ x))
63, 5mpbir 134 . 2 {x A ∣ Ind x}
7 bj-indsuc 9317 . . . . . 6 (Ind x → (y x → suc y x))
87a2i 11 . . . . 5 ((Ind xy x) → (Ind x → suc y x))
98ralimi 2378 . . . 4 (x A (Ind xy x) → x A (Ind x → suc y x))
10 vex 2554 . . . . 5 y V
1110elintrab 3618 . . . 4 (y {x A ∣ Ind x} ↔ x A (Ind xy x))
1210bj-sucex 9308 . . . . 5 suc y V
1312elintrab 3618 . . . 4 (suc y {x A ∣ Ind x} ↔ x A (Ind x → suc y x))
149, 11, 133imtr4i 190 . . 3 (y {x A ∣ Ind x} → suc y {x A ∣ Ind x})
1514rgen 2368 . 2 y {x A ∣ Ind x}suc y {x A ∣ Ind x}
16 df-bj-ind 9316 . 2 (Ind {x A ∣ Ind x} ↔ (∅ {x A ∣ Ind x} y {x A ∣ Ind x}suc y {x A ∣ Ind x}))
176, 15, 16mpbir2an 848 1 Ind {x A ∣ Ind x}
Colors of variables: wff set class
Syntax hints:  wi 4   wcel 1390  wral 2300  {crab 2304  c0 3218   cint 3606  suc csuc 4068  Ind wind 9315
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 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-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-nul 3874  ax-pr 3935  ax-un 4136  ax-bd0 9202  ax-bdor 9205  ax-bdex 9208  ax-bdeq 9209  ax-bdel 9210  ax-bdsb 9211  ax-bdsep 9273
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-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-nul 3219  df-sn 3373  df-pr 3374  df-uni 3572  df-int 3607  df-suc 4074  df-bdc 9230  df-bj-ind 9316
This theorem is referenced by:  bj-omind  9322
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