Theorem onmindif2 2313

Description: The minimum of a class of ordinal numbers is less than the minimum of that class with its minimum removed.

Assertion

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

Proof of Theorem onmindif2

Step	Hyp	Ref	Expression
1		onnmin 2270	. . . . . . . . . . 11 ⊢ ((A ⊆ On ∧ x ∈ A) → ¬ x ∈ ∩A)
2	1	adantlr 310	. . . . . . . . . 10 ⊢ (((A ⊆ On ∧ ¬ A = ∅) ∧ x ∈ A) → ¬ x ∈ ∩A)
3		ontri1 2232	. . . . . . . . . . . 12 ⊢ ((∩A ∈ On ∧ x ∈ On) → (∩A ⊆ x ↔ ¬ x ∈ ∩A))
4		onsseleq 2254	. . . . . . . . . . . 12 ⊢ ((∩A ∈ On ∧ x ∈ On) → (∩A ⊆ x ↔ (∩A ∈ x ∨ ∩A = x)))
5	3, 4	bitr3d 408	. . . . . . . . . . 11 ⊢ ((∩A ∈ On ∧ x ∈ On) → (¬ x ∈ ∩A ↔ (∩A ∈ x ∨ ∩A = x)))
6		oninton 2267	. . . . . . . . . . . 12 ⊢ ((A ⊆ On ∧ ¬ A = ∅) → ∩A ∈ On)
7	6	adantr 306	. . . . . . . . . . 11 ⊢ (((A ⊆ On ∧ ¬ A = ∅) ∧ x ∈ A) → ∩A ∈ On)
8		ssel2 1503	. . . . . . . . . . . 12 ⊢ ((A ⊆ On ∧ x ∈ A) → x ∈ On)
9	8	adantlr 310	. . . . . . . . . . 11 ⊢ (((A ⊆ On ∧ ¬ A = ∅) ∧ x ∈ A) → x ∈ On)
10	5, 7, 9	sylanc 361	. . . . . . . . . 10 ⊢ (((A ⊆ On ∧ ¬ A = ∅) ∧ x ∈ A) → (¬ x ∈ ∩A ↔ (∩A ∈ x ∨ ∩A = x)))
11	2, 10	mpbid 170	. . . . . . . . 9 ⊢ (((A ⊆ On ∧ ¬ A = ∅) ∧ x ∈ A) → (∩A ∈ x ∨ ∩A = x))
12	11	ord 202	. . . . . . . 8 ⊢ (((A ⊆ On ∧ ¬ A = ∅) ∧ x ∈ A) → (¬ ∩A ∈ x → ∩A = x))
13		elsn 1820	. . . . . . . . 9 ⊢ (x ∈ {∩A} ↔ x = ∩A)
14		cleqcom 1103	. . . . . . . . 9 ⊢ (x = ∩A ↔ ∩A = x)
15	13, 14	bitr 151	. . . . . . . 8 ⊢ (x ∈ {∩A} ↔ ∩A = x)
16	12, 15	syl6ibr 186	. . . . . . 7 ⊢ (((A ⊆ On ∧ ¬ A = ∅) ∧ x ∈ A) → (¬ ∩A ∈ x → x ∈ {∩A}))
17	16	con1d 85	. . . . . 6 ⊢ (((A ⊆ On ∧ ¬ A = ∅) ∧ x ∈ A) → (¬ x ∈ {∩A} → ∩A ∈ x))
18	17	exp 291	. . . . 5 ⊢ ((A ⊆ On ∧ ¬ A = ∅) → (x ∈ A → (¬ x ∈ {∩A} → ∩A ∈ x)))
19	18	imp3a 279	. . . 4 ⊢ ((A ⊆ On ∧ ¬ A = ∅) → ((x ∈ A ∧ ¬ x ∈ {∩A}) → ∩A ∈ x))
20		eldif 1496	. . . 4 ⊢ (x ∈ (A ∖ {∩A}) ↔ (x ∈ A ∧ ¬ x ∈ {∩A}))
21	19, 20	syl5ib 181	. . 3 ⊢ ((A ⊆ On ∧ ¬ A = ∅) → (x ∈ (A ∖ {∩A}) → ∩A ∈ x))
22	21	r19.21aiv 1259	. 2 ⊢ ((A ⊆ On ∧ ¬ A = ∅) → ∀x ∈ (A ∖ {∩A})∩A ∈ x)
23		intex 1986	. . . 4 ⊢ (¬ A = ∅ ↔ ∩A ∈ V)
24		elintg 1973	. . . 4 ⊢ (∩A ∈ V → (∩A ∈ ∩ (A ∖ {∩A}) ↔ ∀x ∈ (A ∖ {∩A})∩A ∈ x))
25	23, 24	sylbi 174	. . 3 ⊢ (¬ A = ∅ → (∩A ∈ ∩ (A ∖ {∩A}) ↔ ∀x ∈ (A ∖ {∩A})∩A ∈ x))
26	25	adantl 305	. 2 ⊢ ((A ⊆ On ∧ ¬ A = ∅) → (∩A ∈ ∩ (A ∖ {∩A}) ↔ ∀x ∈ (A ∖ {∩A})∩A ∈ x))
27	22, 26	mpbird 171	1 ⊢ ((A ⊆ On ∧ ¬ A = ∅) → ∩A ∈ ∩ (A ∖ {∩A}))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∀wral 1201  Vcvv 1348   ∖ cdif 1484   ⊆ wss 1487  ∅c0 1707  {csn 1808  ∩cint 1965  Oncon0 2199

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203

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