Theorem oneqmini 6418

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 4931	. . . . . 6 ⊢ (𝐴 ⊆ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ⊆ 𝑥)
2		ssel 3932	. . . . . . . . . . . 12 ⊢ (𝐵 ⊆ On → (𝐴 ∈ 𝐵 → 𝐴 ∈ On))
3		ssel 3932	. . . . . . . . . . . 12 ⊢ (𝐵 ⊆ On → (𝑥 ∈ 𝐵 → 𝑥 ∈ On))
4	2, 3	anim12d 621	. . . . . . . . . . 11 ⊢ (𝐵 ⊆ On → ((𝐴 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝐴 ∈ On ∧ 𝑥 ∈ On)))
5		ontri1 6399	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝐴))
6	4, 5	syl6 36	. . . . . . . . . 10 ⊢ (𝐵 ⊆ On → ((𝐴 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝐴 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝐴)))
7	6	expdimp 458	. . . . . . . . 9 ⊢ ((𝐵 ⊆ On ∧ 𝐴 ∈ 𝐵) → (𝑥 ∈ 𝐵 → (𝐴 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝐴)))
8	7	pm5.74d 276	. . . . . . . 8 ⊢ ((𝐵 ⊆ On ∧ 𝐴 ∈ 𝐵) → ((𝑥 ∈ 𝐵 → 𝐴 ⊆ 𝑥) ↔ (𝑥 ∈ 𝐵 → ¬ 𝑥 ∈ 𝐴)))
9		con2b 362	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 → ¬ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵))
10	8, 9	bitrdi 290	. . . . . . 7 ⊢ ((𝐵 ⊆ On ∧ 𝐴 ∈ 𝐵) → ((𝑥 ∈ 𝐵 → 𝐴 ⊆ 𝑥) ↔ (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵)))
11	10	ralbidv2 3186	. . . . . 6 ⊢ ((𝐵 ⊆ On ∧ 𝐴 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 𝐴 ⊆ 𝑥 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵))
12	1, 11	bitrid 286	. . . . 5 ⊢ ((𝐵 ⊆ On ∧ 𝐴 ∈ 𝐵) → (𝐴 ⊆ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵))
13	12	biimprd 251	. . . 4 ⊢ ((𝐵 ⊆ On ∧ 𝐴 ∈ 𝐵) → (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵 → 𝐴 ⊆ ∩ 𝐵))
14	13	expimpd 459	. . 3 ⊢ (𝐵 ⊆ On → ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) → 𝐴 ⊆ ∩ 𝐵))
15		intss1 4930	. . . . 5 ⊢ (𝐴 ∈ 𝐵 → ∩ 𝐵 ⊆ 𝐴)
16	15	a1i 11	. . . 4 ⊢ (𝐵 ⊆ On → (𝐴 ∈ 𝐵 → ∩ 𝐵 ⊆ 𝐴))
17	16	adantrd 497	. . 3 ⊢ (𝐵 ⊆ On → ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) → ∩ 𝐵 ⊆ 𝐴))
18	14, 17	jcad 522	. 2 ⊢ (𝐵 ⊆ On → ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) → (𝐴 ⊆ ∩ 𝐵 ∧ ∩ 𝐵 ⊆ 𝐴)))
19		eqss 3953	. 2 ⊢ (𝐴 = ∩ 𝐵 ↔ (𝐴 ⊆ ∩ 𝐵 ∧ ∩ 𝐵 ⊆ 𝐴))
20	18, 19	imbitrrdi 255	1 ⊢ (𝐵 ⊆ On → ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) → 𝐴 = ∩ 𝐵))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081   ⊆ wss 3906  ∩ cint 4914  Oncon0 6364

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-11 2195  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-br 5112  df-opab 5176  df-tr 5221  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-ord 6367  df-on 6368

This theorem is used by:  oneqmin  7805  alephval3  10110  cfsuc  10256  alephval2  10572