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Theorem bj-indint 11255
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 11251 . . . . 5 (Ind 𝑥 ↔ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥))
21simplbi 268 . . . 4 (Ind 𝑥 → ∅ ∈ 𝑥)
32rgenw 2426 . . 3 𝑥𝐴 (Ind 𝑥 → ∅ ∈ 𝑥)
4 0ex 3939 . . . 4 ∅ ∈ V
54elintrab 3682 . . 3 (∅ ∈ {𝑥𝐴 ∣ Ind 𝑥} ↔ ∀𝑥𝐴 (Ind 𝑥 → ∅ ∈ 𝑥))
63, 5mpbir 144 . 2 ∅ ∈ {𝑥𝐴 ∣ Ind 𝑥}
7 bj-indsuc 11252 . . . . . 6 (Ind 𝑥 → (𝑦𝑥 → suc 𝑦𝑥))
87a2i 11 . . . . 5 ((Ind 𝑥𝑦𝑥) → (Ind 𝑥 → suc 𝑦𝑥))
98ralimi 2434 . . . 4 (∀𝑥𝐴 (Ind 𝑥𝑦𝑥) → ∀𝑥𝐴 (Ind 𝑥 → suc 𝑦𝑥))
10 vex 2618 . . . . 5 𝑦 ∈ V
1110elintrab 3682 . . . 4 (𝑦 {𝑥𝐴 ∣ Ind 𝑥} ↔ ∀𝑥𝐴 (Ind 𝑥𝑦𝑥))
1210bj-sucex 11243 . . . . 5 suc 𝑦 ∈ V
1312elintrab 3682 . . . 4 (suc 𝑦 {𝑥𝐴 ∣ Ind 𝑥} ↔ ∀𝑥𝐴 (Ind 𝑥 → suc 𝑦𝑥))
149, 11, 133imtr4i 199 . . 3 (𝑦 {𝑥𝐴 ∣ Ind 𝑥} → suc 𝑦 {𝑥𝐴 ∣ Ind 𝑥})
1514rgen 2424 . 2 𝑦 {𝑥𝐴 ∣ Ind 𝑥}suc 𝑦 {𝑥𝐴 ∣ Ind 𝑥}
16 df-bj-ind 11251 . 2 (Ind {𝑥𝐴 ∣ Ind 𝑥} ↔ (∅ ∈ {𝑥𝐴 ∣ Ind 𝑥} ∧ ∀𝑦 {𝑥𝐴 ∣ Ind 𝑥}suc 𝑦 {𝑥𝐴 ∣ Ind 𝑥}))
176, 15, 16mpbir2an 886 1 Ind {𝑥𝐴 ∣ Ind 𝑥}
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1436  wral 2355  {crab 2359  c0 3275   cint 3670  suc csuc 4164  Ind wind 11250
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-nul 3938  ax-pr 4008  ax-un 4232  ax-bd0 11133  ax-bdor 11136  ax-bdex 11139  ax-bdeq 11140  ax-bdel 11141  ax-bdsb 11142  ax-bdsep 11204
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2617  df-dif 2990  df-un 2992  df-nul 3276  df-sn 3436  df-pr 3437  df-uni 3636  df-int 3671  df-suc 4170  df-bdc 11161  df-bj-ind 11251
This theorem is referenced by:  bj-omind  11258
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