Theorem onmindif2 7855

Description: The minimum of a class of ordinal numbers is less than the minimum of that class with its minimum removed. (Contributed by NM, 20-Nov-2003.)

Assertion

Ref	Expression
onmindif2	⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ ∩ (𝐴 ∖ {∩ 𝐴}))

Proof of Theorem onmindif2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldifsn 4750	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ {∩ 𝐴}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ≠ ∩ 𝐴))
2		onnmin 7846	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 ∈ ∩ 𝐴)
3	2	adantlr 728	. . . . . . . . 9 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 ∈ ∩ 𝐴)
4		oninton 7843	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ On)
5		ssel2 3908	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On)
6	5	adantlr 728	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On)
7		ontri1 6423	. . . . . . . . . . 11 ⊢ ((∩ 𝐴 ∈ On ∧ 𝑥 ∈ On) → (∩ 𝐴 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ ∩ 𝐴))
8		onsseleq 6430	. . . . . . . . . . 11 ⊢ ((∩ 𝐴 ∈ On ∧ 𝑥 ∈ On) → (∩ 𝐴 ⊆ 𝑥 ↔ (∩ 𝐴 ∈ 𝑥 ∨ ∩ 𝐴 = 𝑥)))
9	7, 8	bitr3d 284	. . . . . . . . . 10 ⊢ ((∩ 𝐴 ∈ On ∧ 𝑥 ∈ On) → (¬ 𝑥 ∈ ∩ 𝐴 ↔ (∩ 𝐴 ∈ 𝑥 ∨ ∩ 𝐴 = 𝑥)))
10	4, 6, 9	syl2an2r 698	. . . . . . . . 9 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝐴) → (¬ 𝑥 ∈ ∩ 𝐴 ↔ (∩ 𝐴 ∈ 𝑥 ∨ ∩ 𝐴 = 𝑥)))
11	3, 10	mpbid 235	. . . . . . . 8 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝐴) → (∩ 𝐴 ∈ 𝑥 ∨ ∩ 𝐴 = 𝑥))
12	11	ord 878	. . . . . . 7 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝐴) → (¬ ∩ 𝐴 ∈ 𝑥 → ∩ 𝐴 = 𝑥))
13		eqcom 2770	. . . . . . 7 ⊢ (∩ 𝐴 = 𝑥 ↔ 𝑥 = ∩ 𝐴)
14	12, 13	imbitrdi 254	. . . . . 6 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝐴) → (¬ ∩ 𝐴 ∈ 𝑥 → 𝑥 = ∩ 𝐴))
15	14	necon1ad 2976	. . . . 5 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝐴) → (𝑥 ≠ ∩ 𝐴 → ∩ 𝐴 ∈ 𝑥))
16	15	expimpd 459	. . . 4 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ((𝑥 ∈ 𝐴 ∧ 𝑥 ≠ ∩ 𝐴) → ∩ 𝐴 ∈ 𝑥))
17	1, 16	biimtrid 245	. . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → (𝑥 ∈ (𝐴 ∖ {∩ 𝐴}) → ∩ 𝐴 ∈ 𝑥))
18	17	ralrimiv 3157	. 2 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∀𝑥 ∈ (𝐴 ∖ {∩ 𝐴})∩ 𝐴 ∈ 𝑥)
19		intex 5312	. . . 4 ⊢ (𝐴 ≠ ∅ ↔ ∩ 𝐴 ∈ V)
20		elintg 4920	. . . 4 ⊢ (∩ 𝐴 ∈ V → (∩ 𝐴 ∈ ∩ (𝐴 ∖ {∩ 𝐴}) ↔ ∀𝑥 ∈ (𝐴 ∖ {∩ 𝐴})∩ 𝐴 ∈ 𝑥))
21	19, 20	sylbi 220	. . 3 ⊢ (𝐴 ≠ ∅ → (∩ 𝐴 ∈ ∩ (𝐴 ∖ {∩ 𝐴}) ↔ ∀𝑥 ∈ (𝐴 ∖ {∩ 𝐴})∩ 𝐴 ∈ 𝑥))
22	21	adantl 487	. 2 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → (∩ 𝐴 ∈ ∩ (𝐴 ∖ {∩ 𝐴}) ↔ ∀𝑥 ∈ (𝐴 ∖ {∩ 𝐴})∩ 𝐴 ∈ 𝑥))
23	18, 22	mpbird 260	1 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ ∩ (𝐴 ∖ {∩ 𝐴}))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080  Vcvv 3454   ⊆ wss 3900   ∖ cdif 4048  ∅c0 4281  {csn 4588  ∩ 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-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)