Theorem bm2.5ii 4618

Description: Problem 2.5(ii) of [BellMachover] p. 471. (Contributed by NM, 20-Sep-2003.)

Hypothesis

Ref	Expression
bm2.5ii.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
bm2.5ii	⊢ (𝐴 ⊆ On → ∪ 𝐴 = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥})

Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem bm2.5ii

Step	Hyp	Ref	Expression
1		bm2.5ii.1	. . 3 ⊢ 𝐴 ∈ V
2	1	ssonunii 4611	. 2 ⊢ (𝐴 ⊆ On → ∪ 𝐴 ∈ On)
3		intmin 3969	. . 3 ⊢ (∪ 𝐴 ∈ On → ∩ {𝑥 ∈ On ∣ ∪ 𝐴 ⊆ 𝑥} = ∪ 𝐴)
4		unissb 3944	. . . . . 6 ⊢ (∪ 𝐴 ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥)
5	4	a1i 9	. . . . 5 ⊢ (𝑥 ∈ On → (∪ 𝐴 ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥))
6	5	rabbiia 2799	. . . 4 ⊢ {𝑥 ∈ On ∣ ∪ 𝐴 ⊆ 𝑥} = {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥}
7	6	inteqi 3953	. . 3 ⊢ ∩ {𝑥 ∈ On ∣ ∪ 𝐴 ⊆ 𝑥} = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥}
8	3, 7	eqtr3di 2280	. 2 ⊢ (∪ 𝐴 ∈ On → ∪ 𝐴 = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥})
9	2, 8	syl 14	1 ⊢ (𝐴 ⊆ On → ∪ 𝐴 = ∩ {𝑥 ∈ On ∣ ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥})

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1398   ∈ wcel 2203  ∀wral 2520  {crab 2524  Vcvv 2813   ⊆ wss 3211  ∪ cuni 3914  ∩ cint 3949  Oncon0 4484

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-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-in 3217  df-ss 3224  df-uni 3915  df-int 3950  df-tr 4209  df-iord 4487  df-on 4489

This theorem is referenced by: (None)