Theorem oneqmini 6415

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 4933	. . . . . 6 ⊢ (𝐴 ⊆ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐴 ⊆ 𝑥)
2		ssel 3939	. . . . . . . . . . . 12 ⊢ (𝐵 ⊆ On → (𝐴 ∈ 𝐵 → 𝐴 ∈ On))
3		ssel 3939	. . . . . . . . . . . 12 ⊢ (𝐵 ⊆ On → (𝑥 ∈ 𝐵 → 𝑥 ∈ On))
4	2, 3	anim12d 620	. . . . . . . . . . 11 ⊢ (𝐵 ⊆ On → ((𝐴 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝐴 ∈ On ∧ 𝑥 ∈ On)))
5		ontri1 6396	. . . . . . . . . . 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 457	. . . . . . . . 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 3190	. . . . . 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 458	. . 3 ⊢ (𝐵 ⊆ On → ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) → 𝐴 ⊆ ∩ 𝐵))
15		intss1 4932	. . . . 5 ⊢ (𝐴 ∈ 𝐵 → ∩ 𝐵 ⊆ 𝐴)
16	15	a1i 11	. . . 4 ⊢ (𝐵 ⊆ On → (𝐴 ∈ 𝐵 → ∩ 𝐵 ⊆ 𝐴))
17	16	adantrd 496	. . 3 ⊢ (𝐵 ⊆ On → ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) → ∩ 𝐵 ⊆ 𝐴))
18	14, 17	jcad 521	. 2 ⊢ (𝐵 ⊆ On → ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) → (𝐴 ⊆ ∩ 𝐵 ∧ ∩ 𝐵 ⊆ 𝐴)))
19		eqss 3960	. 2 ⊢ (𝐴 = ∩ 𝐵 ↔ (𝐴 ⊆ ∩ 𝐵 ∧ ∩ 𝐵 ⊆ 𝐴))
20	18, 19	imbitrrdi 255	1 ⊢ (𝐵 ⊆ On → ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) → 𝐴 = ∩ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085   ⊆ wss 3913  ∩ cint 4916  Oncon0 6361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-11 2198  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-br 5114  df-opab 5178  df-tr 5223  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-ord 6364  df-on 6365

This theorem is referenced by:  oneqmin  7798  alephval3  10093  cfsuc  10240  alephval2  10556