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Theorem bj-indint 10055
 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 10051 . . . . 5 (Ind 𝑥 ↔ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥))
21simplbi 259 . . . 4 (Ind 𝑥 → ∅ ∈ 𝑥)
32rgenw 2376 . . 3 𝑥𝐴 (Ind 𝑥 → ∅ ∈ 𝑥)
4 0ex 3884 . . . 4 ∅ ∈ V
54elintrab 3627 . . 3 (∅ ∈ {𝑥𝐴 ∣ Ind 𝑥} ↔ ∀𝑥𝐴 (Ind 𝑥 → ∅ ∈ 𝑥))
63, 5mpbir 134 . 2 ∅ ∈ {𝑥𝐴 ∣ Ind 𝑥}
7 bj-indsuc 10052 . . . . . 6 (Ind 𝑥 → (𝑦𝑥 → suc 𝑦𝑥))
87a2i 11 . . . . 5 ((Ind 𝑥𝑦𝑥) → (Ind 𝑥 → suc 𝑦𝑥))
98ralimi 2384 . . . 4 (∀𝑥𝐴 (Ind 𝑥𝑦𝑥) → ∀𝑥𝐴 (Ind 𝑥 → suc 𝑦𝑥))
10 vex 2560 . . . . 5 𝑦 ∈ V
1110elintrab 3627 . . . 4 (𝑦 {𝑥𝐴 ∣ Ind 𝑥} ↔ ∀𝑥𝐴 (Ind 𝑥𝑦𝑥))
1210bj-sucex 10043 . . . . 5 suc 𝑦 ∈ V
1312elintrab 3627 . . . 4 (suc 𝑦 {𝑥𝐴 ∣ Ind 𝑥} ↔ ∀𝑥𝐴 (Ind 𝑥 → suc 𝑦𝑥))
149, 11, 133imtr4i 190 . . 3 (𝑦 {𝑥𝐴 ∣ Ind 𝑥} → suc 𝑦 {𝑥𝐴 ∣ Ind 𝑥})
1514rgen 2374 . 2 𝑦 {𝑥𝐴 ∣ Ind 𝑥}suc 𝑦 {𝑥𝐴 ∣ Ind 𝑥}
16 df-bj-ind 10051 . 2 (Ind {𝑥𝐴 ∣ Ind 𝑥} ↔ (∅ ∈ {𝑥𝐴 ∣ Ind 𝑥} ∧ ∀𝑦 {𝑥𝐴 ∣ Ind 𝑥}suc 𝑦 {𝑥𝐴 ∣ Ind 𝑥}))
176, 15, 16mpbir2an 849 1 Ind {𝑥𝐴 ∣ Ind 𝑥}
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1393  ∀wral 2306  {crab 2310  ∅c0 3224  ∩ cint 3615  suc csuc 4102  Ind wind 10050 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 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-nul 3883  ax-pr 3944  ax-un 4170  ax-bd0 9933  ax-bdor 9936  ax-bdex 9939  ax-bdeq 9940  ax-bdel 9941  ax-bdsb 9942  ax-bdsep 10004 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-nul 3225  df-sn 3381  df-pr 3382  df-uni 3581  df-int 3616  df-suc 4108  df-bdc 9961  df-bj-ind 10051 This theorem is referenced by:  bj-omind  10058
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