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Theorem bj-indint 13300
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 13296 . . . . 5 (Ind 𝑥 ↔ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥))
21simplbi 272 . . . 4 (Ind 𝑥 → ∅ ∈ 𝑥)
32rgenw 2490 . . 3 𝑥𝐴 (Ind 𝑥 → ∅ ∈ 𝑥)
4 0ex 4063 . . . 4 ∅ ∈ V
54elintrab 3791 . . 3 (∅ ∈ {𝑥𝐴 ∣ Ind 𝑥} ↔ ∀𝑥𝐴 (Ind 𝑥 → ∅ ∈ 𝑥))
63, 5mpbir 145 . 2 ∅ ∈ {𝑥𝐴 ∣ Ind 𝑥}
7 bj-indsuc 13297 . . . . . 6 (Ind 𝑥 → (𝑦𝑥 → suc 𝑦𝑥))
87a2i 11 . . . . 5 ((Ind 𝑥𝑦𝑥) → (Ind 𝑥 → suc 𝑦𝑥))
98ralimi 2498 . . . 4 (∀𝑥𝐴 (Ind 𝑥𝑦𝑥) → ∀𝑥𝐴 (Ind 𝑥 → suc 𝑦𝑥))
10 vex 2692 . . . . 5 𝑦 ∈ V
1110elintrab 3791 . . . 4 (𝑦 {𝑥𝐴 ∣ Ind 𝑥} ↔ ∀𝑥𝐴 (Ind 𝑥𝑦𝑥))
1210bj-sucex 13292 . . . . 5 suc 𝑦 ∈ V
1312elintrab 3791 . . . 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 2488 . 2 𝑦 {𝑥𝐴 ∣ Ind 𝑥}suc 𝑦 {𝑥𝐴 ∣ Ind 𝑥}
16 df-bj-ind 13296 . 2 (Ind {𝑥𝐴 ∣ Ind 𝑥} ↔ (∅ ∈ {𝑥𝐴 ∣ Ind 𝑥} ∧ ∀𝑦 {𝑥𝐴 ∣ Ind 𝑥}suc 𝑦 {𝑥𝐴 ∣ Ind 𝑥}))
176, 15, 16mpbir2an 927 1 Ind {𝑥𝐴 ∣ Ind 𝑥}
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1481  wral 2417  {crab 2421  c0 3368   cint 3779  suc csuc 4295  Ind wind 13295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-nul 4062  ax-pr 4139  ax-un 4363  ax-bd0 13182  ax-bdor 13185  ax-bdex 13188  ax-bdeq 13189  ax-bdel 13190  ax-bdsb 13191  ax-bdsep 13253
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-dif 3078  df-un 3080  df-nul 3369  df-sn 3538  df-pr 3539  df-uni 3745  df-int 3780  df-suc 4301  df-bdc 13210  df-bj-ind 13296
This theorem is referenced by:  bj-omind  13303
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