Theorem oneqmin 7781

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 7771	. . . 4 ⊢ ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → ∩ 𝐵 ∈ 𝐵)
2		eleq1 2813	. . . 4 ⊢ (𝐴 = ∩ 𝐵 → (𝐴 ∈ 𝐵 ↔ ∩ 𝐵 ∈ 𝐵))
3	1, 2	syl5ibrcom 246	. . 3 ⊢ ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → (𝐴 = ∩ 𝐵 → 𝐴 ∈ 𝐵))
4		eleq2 2814	. . . . . . 7 ⊢ (𝐴 = ∩ 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∩ 𝐵))
5	4	biimpd 228	. . . . . 6 ⊢ (𝐴 = ∩ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ ∩ 𝐵))
6		onnmin 7779	. . . . . . . 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 3144	. . . 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 6406	. . 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 211	1 ⊢ ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → (𝐴 = ∩ 𝐵 ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098   ≠ wne 2932  ∀wral 3053   ⊆ wss 3940  ∅c0 4314  ∩ cint 4940  Oncon0 6354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-11 2146  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-br 5139  df-opab 5201  df-tr 5256  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-ord 6357  df-on 6358

This theorem is referenced by: (None)