Theorem oneqmini 5693

Description: A way to show that an ordinal number equals the minimum of a collection of ordinal numbers: it must be in the collection, and it must not be larger than any member of the collection. (Contributed by NM, 14-Nov-2003.)

Assertion

Ref	Expression
oneqmini	⊢ (𝐵 ⊆ On → ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) → 𝐴 = ∩ 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem oneqmini

Step	Hyp	Ref	Expression
1		ssint 4428	. . . . . 6 ⊢ (𝐴 ⊆ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ⊆ 𝑥)
2		ssel 3562	. . . . . . . . . . . 12 ⊢ (𝐵 ⊆ On → (𝐴 ∈ 𝐵 → 𝐴 ∈ On))
3		ssel 3562	. . . . . . . . . . . 12 ⊢ (𝐵 ⊆ On → (𝑥 ∈ 𝐵 → 𝑥 ∈ On))
4	2, 3	anim12d 584	. . . . . . . . . . 11 ⊢ (𝐵 ⊆ On → ((𝐴 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝐴 ∈ On ∧ 𝑥 ∈ On)))
5		ontri1 5674	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝐴))
6	4, 5	syl6 34	. . . . . . . . . 10 ⊢ (𝐵 ⊆ On → ((𝐴 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝐴 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝐴)))
7	6	expdimp 452	. . . . . . . . 9 ⊢ ((𝐵 ⊆ On ∧ 𝐴 ∈ 𝐵) → (𝑥 ∈ 𝐵 → (𝐴 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝐴)))
8	7	pm5.74d 261	. . . . . . . 8 ⊢ ((𝐵 ⊆ On ∧ 𝐴 ∈ 𝐵) → ((𝑥 ∈ 𝐵 → 𝐴 ⊆ 𝑥) ↔ (𝑥 ∈ 𝐵 → ¬ 𝑥 ∈ 𝐴)))
9		con2b 348	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 → ¬ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵))
10	8, 9	syl6bb 275	. . . . . . 7 ⊢ ((𝐵 ⊆ On ∧ 𝐴 ∈ 𝐵) → ((𝑥 ∈ 𝐵 → 𝐴 ⊆ 𝑥) ↔ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵)))
11	10	ralbidv2 2967	. . . . . 6 ⊢ ((𝐵 ⊆ On ∧ 𝐴 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 𝐴 ⊆ 𝑥 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵))
12	1, 11	syl5bb 271	. . . . 5 ⊢ ((𝐵 ⊆ On ∧ 𝐴 ∈ 𝐵) → (𝐴 ⊆ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵))
13	12	biimprd 237	. . . 4 ⊢ ((𝐵 ⊆ On ∧ 𝐴 ∈ 𝐵) → (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 → 𝐴 ⊆ ∩ 𝐵))
14	13	expimpd 627	. . 3 ⊢ (𝐵 ⊆ On → ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) → 𝐴 ⊆ ∩ 𝐵))
15		intss1 4427	. . . . 5 ⊢ (𝐴 ∈ 𝐵 → ∩ 𝐵 ⊆ 𝐴)
16	15	a1i 11	. . . 4 ⊢ (𝐵 ⊆ On → (𝐴 ∈ 𝐵 → ∩ 𝐵 ⊆ 𝐴))
17	16	adantrd 483	. . 3 ⊢ (𝐵 ⊆ On → ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) → ∩ 𝐵 ⊆ 𝐴))
18	14, 17	jcad 554	. 2 ⊢ (𝐵 ⊆ On → ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) → (𝐴 ⊆ ∩ 𝐵 ∧ ∩ 𝐵 ⊆ 𝐴)))
19		eqss 3583	. 2 ⊢ (𝐴 = ∩ 𝐵 ↔ (𝐴 ⊆ ∩ 𝐵 ∧ ∩ 𝐵 ⊆ 𝐴))
20	18, 19	syl6ibr 241	1 ⊢ (𝐵 ⊆ On → ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) → 𝐴 = ∩ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540  ∩ cint 4410  Oncon0 5640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644

This theorem is referenced by:  oneqmin  6897  alephval3  8816  cfsuc  8962  alephval2  9273