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Theorem bj-indint 15169
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 15165 . . . . 5 (Ind 𝑥 ↔ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥))
21simplbi 274 . . . 4 (Ind 𝑥 → ∅ ∈ 𝑥)
32rgenw 2545 . . 3 𝑥𝐴 (Ind 𝑥 → ∅ ∈ 𝑥)
4 0ex 4148 . . . 4 ∅ ∈ V
54elintrab 3874 . . 3 (∅ ∈ {𝑥𝐴 ∣ Ind 𝑥} ↔ ∀𝑥𝐴 (Ind 𝑥 → ∅ ∈ 𝑥))
63, 5mpbir 146 . 2 ∅ ∈ {𝑥𝐴 ∣ Ind 𝑥}
7 bj-indsuc 15166 . . . . . 6 (Ind 𝑥 → (𝑦𝑥 → suc 𝑦𝑥))
87a2i 11 . . . . 5 ((Ind 𝑥𝑦𝑥) → (Ind 𝑥 → suc 𝑦𝑥))
98ralimi 2553 . . . 4 (∀𝑥𝐴 (Ind 𝑥𝑦𝑥) → ∀𝑥𝐴 (Ind 𝑥 → suc 𝑦𝑥))
10 vex 2755 . . . . 5 𝑦 ∈ V
1110elintrab 3874 . . . 4 (𝑦 {𝑥𝐴 ∣ Ind 𝑥} ↔ ∀𝑥𝐴 (Ind 𝑥𝑦𝑥))
1210bj-sucex 15161 . . . . 5 suc 𝑦 ∈ V
1312elintrab 3874 . . . 4 (suc 𝑦 {𝑥𝐴 ∣ Ind 𝑥} ↔ ∀𝑥𝐴 (Ind 𝑥 → suc 𝑦𝑥))
149, 11, 133imtr4i 201 . . 3 (𝑦 {𝑥𝐴 ∣ Ind 𝑥} → suc 𝑦 {𝑥𝐴 ∣ Ind 𝑥})
1514rgen 2543 . 2 𝑦 {𝑥𝐴 ∣ Ind 𝑥}suc 𝑦 {𝑥𝐴 ∣ Ind 𝑥}
16 df-bj-ind 15165 . 2 (Ind {𝑥𝐴 ∣ Ind 𝑥} ↔ (∅ ∈ {𝑥𝐴 ∣ Ind 𝑥} ∧ ∀𝑦 {𝑥𝐴 ∣ Ind 𝑥}suc 𝑦 {𝑥𝐴 ∣ Ind 𝑥}))
176, 15, 16mpbir2an 944 1 Ind {𝑥𝐴 ∣ Ind 𝑥}
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2160  wral 2468  {crab 2472  c0 3437   cint 3862  suc csuc 4386  Ind wind 15164
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-nul 4147  ax-pr 4230  ax-un 4454  ax-bd0 15051  ax-bdor 15054  ax-bdex 15057  ax-bdeq 15058  ax-bdel 15059  ax-bdsb 15060  ax-bdsep 15122
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-nul 3438  df-sn 3616  df-pr 3617  df-uni 3828  df-int 3863  df-suc 4392  df-bdc 15079  df-bj-ind 15165
This theorem is referenced by:  bj-omind  15172
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