Theorem oneqmin 7005

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 6995	. . . 4 ⊢ ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → ∩ 𝐵 ∈ 𝐵)
2		eleq1 2689	. . . 4 ⊢ (𝐴 = ∩ 𝐵 → (𝐴 ∈ 𝐵 ↔ ∩ 𝐵 ∈ 𝐵))
3	1, 2	syl5ibrcom 237	. . 3 ⊢ ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → (𝐴 = ∩ 𝐵 → 𝐴 ∈ 𝐵))
4		eleq2 2690	. . . . . . 7 ⊢ (𝐴 = ∩ 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∩ 𝐵))
5	4	biimpd 219	. . . . . 6 ⊢ (𝐴 = ∩ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ ∩ 𝐵))
6		onnmin 7003	. . . . . . . 8 ⊢ ((𝐵 ⊆ On ∧ 𝑥 ∈ 𝐵) → ¬ 𝑥 ∈ ∩ 𝐵)
7	6	ex 450	. . . . . . 7 ⊢ (𝐵 ⊆ On → (𝑥 ∈ 𝐵 → ¬ 𝑥 ∈ ∩ 𝐵))
8	7	con2d 129	. . . . . 6 ⊢ (𝐵 ⊆ On → (𝑥 ∈ ∩ 𝐵 → ¬ 𝑥 ∈ 𝐵))
9	5, 8	syl9r 78	. . . . 5 ⊢ (𝐵 ⊆ On → (𝐴 = ∩ 𝐵 → (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵)))
10	9	ralrimdv 2968	. . . 4 ⊢ (𝐵 ⊆ On → (𝐴 = ∩ 𝐵 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵))
11	10	adantr 481	. . 3 ⊢ ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → (𝐴 = ∩ 𝐵 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵))
12	3, 11	jcad 555	. 2 ⊢ ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → (𝐴 = ∩ 𝐵 → (𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)))
13		oneqmini 5776	. . 3 ⊢ (𝐵 ⊆ On → ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) → 𝐴 = ∩ 𝐵))
14	13	adantr 481	. 2 ⊢ ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) → 𝐴 = ∩ 𝐵))
15	12, 14	impbid 202	1 ⊢ ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → (𝐴 = ∩ 𝐵 ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912   ⊆ wss 3574  ∅c0 3915  ∩ cint 4475  Oncon0 5723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727

This theorem is referenced by: (None)