Theorem onmindif2 7529

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 4721	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ {∩ 𝐴}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ≠ ∩ 𝐴))
2		onnmin 7520	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 ∈ ∩ 𝐴)
3	2	adantlr 713	. . . . . . . . 9 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 ∈ ∩ 𝐴)
4		oninton 7517	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ On)
5		ssel2 3964	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On)
6	5	adantlr 713	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On)
7		ontri1 6227	. . . . . . . . . . 11 ⊢ ((∩ 𝐴 ∈ On ∧ 𝑥 ∈ On) → (∩ 𝐴 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ ∩ 𝐴))
8		onsseleq 6234	. . . . . . . . . . 11 ⊢ ((∩ 𝐴 ∈ On ∧ 𝑥 ∈ On) → (∩ 𝐴 ⊆ 𝑥 ↔ (∩ 𝐴 ∈ 𝑥 ∨ ∩ 𝐴 = 𝑥)))
9	7, 8	bitr3d 283	. . . . . . . . . 10 ⊢ ((∩ 𝐴 ∈ On ∧ 𝑥 ∈ On) → (¬ 𝑥 ∈ ∩ 𝐴 ↔ (∩ 𝐴 ∈ 𝑥 ∨ ∩ 𝐴 = 𝑥)))
10	4, 6, 9	syl2an2r 683	. . . . . . . . 9 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝐴) → (¬ 𝑥 ∈ ∩ 𝐴 ↔ (∩ 𝐴 ∈ 𝑥 ∨ ∩ 𝐴 = 𝑥)))
11	3, 10	mpbid 234	. . . . . . . 8 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝐴) → (∩ 𝐴 ∈ 𝑥 ∨ ∩ 𝐴 = 𝑥))
12	11	ord 860	. . . . . . 7 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝐴) → (¬ ∩ 𝐴 ∈ 𝑥 → ∩ 𝐴 = 𝑥))
13		eqcom 2830	. . . . . . 7 ⊢ (∩ 𝐴 = 𝑥 ↔ 𝑥 = ∩ 𝐴)
14	12, 13	syl6ib 253	. . . . . 6 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝐴) → (¬ ∩ 𝐴 ∈ 𝑥 → 𝑥 = ∩ 𝐴))
15	14	necon1ad 3035	. . . . 5 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) ∧ 𝑥 ∈ 𝐴) → (𝑥 ≠ ∩ 𝐴 → ∩ 𝐴 ∈ 𝑥))
16	15	expimpd 456	. . . 4 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ((𝑥 ∈ 𝐴 ∧ 𝑥 ≠ ∩ 𝐴) → ∩ 𝐴 ∈ 𝑥))
17	1, 16	syl5bi 244	. . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → (𝑥 ∈ (𝐴 ∖ {∩ 𝐴}) → ∩ 𝐴 ∈ 𝑥))
18	17	ralrimiv 3183	. 2 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∀𝑥 ∈ (𝐴 ∖ {∩ 𝐴})∩ 𝐴 ∈ 𝑥)
19		intex 5242	. . . 4 ⊢ (𝐴 ≠ ∅ ↔ ∩ 𝐴 ∈ V)
20		elintg 4886	. . . 4 ⊢ (∩ 𝐴 ∈ V → (∩ 𝐴 ∈ ∩ (𝐴 ∖ {∩ 𝐴}) ↔ ∀𝑥 ∈ (𝐴 ∖ {∩ 𝐴})∩ 𝐴 ∈ 𝑥))
21	19, 20	sylbi 219	. . 3 ⊢ (𝐴 ≠ ∅ → (∩ 𝐴 ∈ ∩ (𝐴 ∖ {∩ 𝐴}) ↔ ∀𝑥 ∈ (𝐴 ∖ {∩ 𝐴})∩ 𝐴 ∈ 𝑥))
22	21	adantl 484	. 2 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → (∩ 𝐴 ∈ ∩ (𝐴 ∖ {∩ 𝐴}) ↔ ∀𝑥 ∈ (𝐴 ∖ {∩ 𝐴})∩ 𝐴 ∈ 𝑥))
23	18, 22	mpbird 259	1 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ ∩ (𝐴 ∖ {∩ 𝐴}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   = wceq 1537   ∈ wcel 2114   ≠ wne 3018  ∀wral 3140  Vcvv 3496   ∖ cdif 3935   ⊆ wss 3938  ∅c0 4293  {csn 4569  ∩ cint 4878  Oncon0 6193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-br 5069  df-opab 5131  df-tr 5175  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-ord 6196  df-on 6197

This theorem is referenced by: (None)