Theorem oneqmin 7848

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 7838	. . . 4 ⊢ ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → ∩ 𝐵 ∈ 𝐵)
2		eleq1 2852	. . . 4 ⊢ (𝐴 = ∩ 𝐵 → (𝐴 ∈ 𝐵 ↔ ∩ 𝐵 ∈ 𝐵))
3	1, 2	syl5ibrcom 250	. . 3 ⊢ ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → (𝐴 = ∩ 𝐵 → 𝐴 ∈ 𝐵))
4		eleq2 2853	. . . . . . 7 ⊢ (𝐴 = ∩ 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∩ 𝐵))
5	4	biimpd 232	. . . . . 6 ⊢ (𝐴 = ∩ 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ ∩ 𝐵))
6		onnmin 7846	. . . . . . . 8 ⊢ ((𝐵 ⊆ On ∧ 𝑥 ∈ 𝐵) → ¬ 𝑥 ∈ ∩ 𝐵)
7	6	ex 418	. . . . . . 7 ⊢ (𝐵 ⊆ On → (𝑥 ∈ 𝐵 → ¬ 𝑥 ∈ ∩ 𝐵))
8	7	con2d 135	. . . . . 6 ⊢ (𝐵 ⊆ On → (𝑥 ∈ ∩ 𝐵 → ¬ 𝑥 ∈ 𝐵))
9	5, 8	syl9r 79	. . . . 5 ⊢ (𝐵 ⊆ On → (𝐴 = ∩ 𝐵 → (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵)))
10	9	ralrimdv 3164	. . . 4 ⊢ (𝐵 ⊆ On → (𝐴 = ∩ 𝐵 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵))
11	10	adantr 486	. . 3 ⊢ ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → (𝐴 = ∩ 𝐵 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵))
12	3, 11	jcad 522	. 2 ⊢ ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → (𝐴 = ∩ 𝐵 → (𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)))
13		oneqmini 6442	. . 3 ⊢ (𝐵 ⊆ On → ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) → 𝐴 = ∩ 𝐵))
14	13	adantr 486	. 2 ⊢ ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵) → 𝐴 = ∩ 𝐵))
15	12, 14	impbid 215	1 ⊢ ((𝐵 ⊆ On ∧ 𝐵 ≠ ∅) → (𝐴 = ∩ 𝐵 ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝐵)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080   ⊆ wss 3900  ∅c0 4281  ∩ cint 4912  Oncon0 6388

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-11 2197  ax-ext 2735  ax-sep 5255  ax-pr 5398

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-br 5109  df-opab 5174  df-tr 5219  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-ord 6391  df-on 6392

This theorem is used by: (None)