Theorem oneqmin 7802

Description: A way to show that an ordinal number equals the minimum of a nonempty 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
oneqmin	⊢ ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → (𝐴 = ∩ 𝐵 ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem oneqmin

Step	Hyp	Ref	Expression
1		onint 7792	. . . 4 ⊢ ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → ∩ 𝐵 ∈ 𝐵)
2		eleq1 2821	. . . 4 ⊢ (𝐴 = ∩ 𝐵 → (𝐴 ∈ 𝐵 ↔ ∩ 𝐵 ∈ 𝐵))
3	1, 2	syl5ibrcom 247	. . 3 ⊢ ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → (𝐴 = ∩ 𝐵 → 𝐴 ∈ 𝐵))
4		eleq2 2822	. . . . . . 7 ⊢ (𝐴 = ∩ 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∩ 𝐵))
5	4	biimpd 229	. . . . . 6 ⊢ (𝐴 = ∩ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ ∩ 𝐵))
6		onnmin 7800	. . . . . . . 8 ⊢ ((𝐵 ⊆ On ∧ 𝑥 ∈ 𝐵) → ¬ 𝑥 ∈ ∩ 𝐵)
7	6	ex 412	. . . . . . 7 ⊢ (𝐵 ⊆ On → (𝑥 ∈ 𝐵 → ¬ 𝑥 ∈ ∩ 𝐵))
8	7	con2d 134	. . . . . 6 ⊢ (𝐵 ⊆ On → (𝑥 ∈ ∩ 𝐵 → ¬ 𝑥 ∈ 𝐵))
9	5, 8	syl9r 78	. . . . 5 ⊢ (𝐵 ⊆ On → (𝐴 = ∩ 𝐵 → (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵)))
10	9	ralrimdv 3139	. . . 4 ⊢ (𝐵 ⊆ On → (𝐴 = ∩ 𝐵 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵))
11	10	adantr 480	. . 3 ⊢ ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → (𝐴 = ∩ 𝐵 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵))
12	3, 11	jcad 512	. 2 ⊢ ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → (𝐴 = ∩ 𝐵 → (𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)))
13		oneqmini 6416	. . 3 ⊢ (𝐵 ⊆ On → ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) → 𝐴 = ∩ 𝐵))
14	13	adantr 480	. 2 ⊢ ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) → 𝐴 = ∩ 𝐵))
15	12, 14	impbid 212	1 ⊢ ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → (𝐴 = ∩ 𝐵 ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107   ≠ wne 2931  ∀wral 3050   ⊆ wss 3931  ∅c0 4313  ∩ cint 4926  Oncon0 6363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-11 2156  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-int 4927  df-br 5124  df-opab 5186  df-tr 5240  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-ord 6366  df-on 6367

This theorem is referenced by: (None)