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 e. |^| (A \ {|^|A}))

Proof of Theorem onmindif2

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

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   \/ wo 195   /\ wa 196   = wceq 1091   e. wcel 1092  A.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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