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Theorem bj-indint 7154
 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 7150 . . . . 5 (Ind x ↔ (∅ x y x suc y x))
21simplbi 259 . . . 4 (Ind x → ∅ x)
32rgenw 2354 . . 3 x A (Ind x → ∅ x)
4 0ex 3858 . . . 4 V
54elintrab 3601 . . 3 (∅ {x A ∣ Ind x} ↔ x A (Ind x → ∅ x))
63, 5mpbir 134 . 2 {x A ∣ Ind x}
7 bj-indsuc 7151 . . . . . 6 (Ind x → (y x → suc y x))
87a2i 11 . . . . 5 ((Ind xy x) → (Ind x → suc y x))
98ralimi 2362 . . . 4 (x A (Ind xy x) → x A (Ind x → suc y x))
10 vex 2538 . . . . 5 y V
1110elintrab 3601 . . . 4 (y {x A ∣ Ind x} ↔ x A (Ind xy x))
1210bj-sucex 7146 . . . . 5 suc y V
1312elintrab 3601 . . . 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 2352 . 2 y {x A ∣ Ind x}suc y {x A ∣ Ind x}
16 df-bj-ind 7150 . 2 (Ind {x A ∣ Ind x} ↔ (∅ {x A ∣ Ind x} y {x A ∣ Ind x}suc y {x A ∣ Ind x}))
176, 15, 16mpbir2an 837 1 Ind {x A ∣ Ind x}
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1374  ∀wral 2284  {crab 2288  ∅c0 3201  ∩ cint 3589  suc csuc 4051  Ind wind 7149 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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-nul 3857  ax-pr 3918  ax-un 4120  ax-bd0 7040  ax-bdor 7043  ax-bdex 7046  ax-bdeq 7047  ax-bdel 7048  ax-bdsb 7049  ax-bdsep 7111 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-rab 2293  df-v 2537  df-dif 2897  df-un 2899  df-nul 3202  df-sn 3356  df-pr 3357  df-uni 3555  df-int 3590  df-suc 4057  df-bdc 7068  df-bj-ind 7150 This theorem is referenced by:  bj-omind  7156
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