Theorem bj-indint 14223

Description: The property of being an inductive class is closed under intersections. (Contributed by BJ, 30-Nov-2019.)

Assertion

Ref	Expression
bj-indint	⊢ Ind ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥}

Distinct variable group:   𝑥,𝐴

Proof of Theorem bj-indint

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-bj-ind 14219	. . . . 5 ⊢ (Ind 𝑥 ↔ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥))
2	1	simplbi 274	. . . 4 ⊢ (Ind 𝑥 → ∅ ∈ 𝑥)
3	2	rgenw 2530	. . 3 ⊢ ∀𝑥 ∈ 𝐴 (Ind 𝑥 → ∅ ∈ 𝑥)
4		0ex 4125	. . . 4 ⊢ ∅ ∈ V
5	4	elintrab 3852	. . 3 ⊢ (∅ ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} ↔ ∀𝑥 ∈ 𝐴 (Ind 𝑥 → ∅ ∈ 𝑥))
6	3, 5	mpbir 146	. 2 ⊢ ∅ ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥}
7		bj-indsuc 14220	. . . . . 6 ⊢ (Ind 𝑥 → (𝑦 ∈ 𝑥 → suc 𝑦 ∈ 𝑥))
8	7	a2i 11	. . . . 5 ⊢ ((Ind 𝑥 → 𝑦 ∈ 𝑥) → (Ind 𝑥 → suc 𝑦 ∈ 𝑥))
9	8	ralimi 2538	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (Ind 𝑥 → 𝑦 ∈ 𝑥) → ∀𝑥 ∈ 𝐴 (Ind 𝑥 → suc 𝑦 ∈ 𝑥))
10		vex 2738	. . . . 5 ⊢ 𝑦 ∈ V
11	10	elintrab 3852	. . . 4 ⊢ (𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} ↔ ∀𝑥 ∈ 𝐴 (Ind 𝑥 → 𝑦 ∈ 𝑥))
12	10	bj-sucex 14215	. . . . 5 ⊢ suc 𝑦 ∈ V
13	12	elintrab 3852	. . . 4 ⊢ (suc 𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} ↔ ∀𝑥 ∈ 𝐴 (Ind 𝑥 → suc 𝑦 ∈ 𝑥))
14	9, 11, 13	3imtr4i 201	. . 3 ⊢ (𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} → suc 𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥})
15	14	rgen 2528	. 2 ⊢ ∀𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥}suc 𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥}
16		df-bj-ind 14219	. 2 ⊢ (Ind ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} ↔ (∅ ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} ∧ ∀𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥}suc 𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥}))
17	6, 15, 16	mpbir2an 942	1 ⊢ Ind ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥}

Syntax hints:   → wi 4   ∈ wcel 2146  ∀wral 2453  {crab 2457  ∅c0 3420  ∩ cint 3840  suc csuc 4359  Ind wind 14218

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-nul 4124  ax-pr 4203  ax-un 4427  ax-bd0 14105  ax-bdor 14108  ax-bdex 14111  ax-bdeq 14112  ax-bdel 14113  ax-bdsb 14114  ax-bdsep 14176

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-dif 3129  df-un 3131  df-nul 3421  df-sn 3595  df-pr 3596  df-uni 3806  df-int 3841  df-suc 4365  df-bdc 14133  df-bj-ind 14219

This theorem is referenced by:  bj-omind  14226