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Theorem bj-indint 12714
Description: The property of being an inductive class is closed under intersections. (Contributed by BJ, 30-Nov-2019.)
Assertion
Ref Expression
bj-indint Ind {𝑥𝐴 ∣ 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 12710 . . . . 5 (Ind 𝑥 ↔ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥))
21simplbi 270 . . . 4 (Ind 𝑥 → ∅ ∈ 𝑥)
32rgenw 2446 . . 3 𝑥𝐴 (Ind 𝑥 → ∅ ∈ 𝑥)
4 0ex 3995 . . . 4 ∅ ∈ V
54elintrab 3730 . . 3 (∅ ∈ {𝑥𝐴 ∣ Ind 𝑥} ↔ ∀𝑥𝐴 (Ind 𝑥 → ∅ ∈ 𝑥))
63, 5mpbir 145 . 2 ∅ ∈ {𝑥𝐴 ∣ Ind 𝑥}
7 bj-indsuc 12711 . . . . . 6 (Ind 𝑥 → (𝑦𝑥 → suc 𝑦𝑥))
87a2i 11 . . . . 5 ((Ind 𝑥𝑦𝑥) → (Ind 𝑥 → suc 𝑦𝑥))
98ralimi 2454 . . . 4 (∀𝑥𝐴 (Ind 𝑥𝑦𝑥) → ∀𝑥𝐴 (Ind 𝑥 → suc 𝑦𝑥))
10 vex 2644 . . . . 5 𝑦 ∈ V
1110elintrab 3730 . . . 4 (𝑦 {𝑥𝐴 ∣ Ind 𝑥} ↔ ∀𝑥𝐴 (Ind 𝑥𝑦𝑥))
1210bj-sucex 12702 . . . . 5 suc 𝑦 ∈ V
1312elintrab 3730 . . . 4 (suc 𝑦 {𝑥𝐴 ∣ Ind 𝑥} ↔ ∀𝑥𝐴 (Ind 𝑥 → suc 𝑦𝑥))
149, 11, 133imtr4i 200 . . 3 (𝑦 {𝑥𝐴 ∣ Ind 𝑥} → suc 𝑦 {𝑥𝐴 ∣ Ind 𝑥})
1514rgen 2444 . 2 𝑦 {𝑥𝐴 ∣ Ind 𝑥}suc 𝑦 {𝑥𝐴 ∣ Ind 𝑥}
16 df-bj-ind 12710 . 2 (Ind {𝑥𝐴 ∣ Ind 𝑥} ↔ (∅ ∈ {𝑥𝐴 ∣ Ind 𝑥} ∧ ∀𝑦 {𝑥𝐴 ∣ Ind 𝑥}suc 𝑦 {𝑥𝐴 ∣ Ind 𝑥}))
176, 15, 16mpbir2an 894 1 Ind {𝑥𝐴 ∣ Ind 𝑥}
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1448  wral 2375  {crab 2379  c0 3310   cint 3718  suc csuc 4225  Ind wind 12709
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-nul 3994  ax-pr 4069  ax-un 4293  ax-bd0 12592  ax-bdor 12595  ax-bdex 12598  ax-bdeq 12599  ax-bdel 12600  ax-bdsb 12601  ax-bdsep 12663
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-dif 3023  df-un 3025  df-nul 3311  df-sn 3480  df-pr 3481  df-uni 3684  df-int 3719  df-suc 4231  df-bdc 12620  df-bj-ind 12710
This theorem is referenced by:  bj-omind  12717
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