Theorem onmindif2 3893

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 C_ On /\ A =/= (/)) -> |^|A e. |^|(A \ {|^|A}))

Proof of Theorem onmindif2

Step	Hyp	Ref	Expression
1		onnmin 3884	. . . . . . . . . 10 |- ((A C_ On /\ x e. A) -> -. x e. |^|A)
2	1	adantlr 429	. . . . . . . . 9 |- (((A C_ On /\ A =/= (/)) /\ x e. A) -> -. x e. |^|A)
3		oninton 3881	. . . . . . . . . . 11 |- ((A C_ On /\ A =/= (/)) -> |^|A e. On)
4	3	adantr 425	. . . . . . . . . 10 |- (((A C_ On /\ A =/= (/)) /\ x e. A) -> |^|A e. On)
5		ssel2 2616	. . . . . . . . . . 11 |- ((A C_ On /\ x e. A) -> x e. On)
6	5	adantlr 429	. . . . . . . . . 10 |- (((A C_ On /\ A =/= (/)) /\ x e. A) -> x e. On)
7		ontri1 3695	. . . . . . . . . . 11 |- ((|^|A e. On /\ x e. On) -> (|^|A C_ x <-> -. x e. |^|A))
8		onsseleq 3704	. . . . . . . . . . 11 |- ((|^|A e. On /\ x e. On) -> (|^|A C_ x <-> (|^|A e. x \/ |^|A = x)))
9	7, 8	bitr3d 589	. . . . . . . . . 10 |- ((|^|A e. On /\ x e. On) -> (-. x e. |^|A <-> (|^|A e. x \/ |^|A = x)))
10	4, 6, 9	syl11anc 524	. . . . . . . . 9 |- (((A C_ On /\ A =/= (/)) /\ x e. A) -> (-. x e. |^|A <-> (|^|A e. x \/ |^|A = x)))
11	2, 10	mpbid 212	. . . . . . . 8 |- (((A C_ On /\ A =/= (/)) /\ x e. A) -> (|^|A e. x \/ |^|A = x))
12	11	ord 249	. . . . . . 7 |- (((A C_ On /\ A =/= (/)) /\ x e. A) -> (-. |^|A e. x -> |^|A = x))
13		eqcom 1886	. . . . . . 7 |- (|^|A = x <-> x = |^|A)
14	12, 13	syl6ib 229	. . . . . 6 |- (((A C_ On /\ A =/= (/)) /\ x e. A) -> (-. |^|A e. x -> x = |^|A))
15	14	necon1ad 2079	. . . . 5 |- (((A C_ On /\ A =/= (/)) /\ x e. A) -> (x =/= |^|A -> |^|A e. x))
16	15	expimpd 404	. . . 4 |- ((A C_ On /\ A =/= (/)) -> ((x e. A /\ x =/= |^|A) -> |^|A e. x))
17		eldifsn 3123	. . . 4 |- (x e. (A \ {|^|A}) <-> (x e. A /\ x =/= |^|A))
18	16, 17	syl5ib 223	. . 3 |- ((A C_ On /\ A =/= (/)) -> (x e. (A \ {|^|A}) -> |^|A e. x))
19	18	r19.21aiv 2175	. 2 |- ((A C_ On /\ A =/= (/)) -> A.x e. (A \ {|^|A})|^|A e. x)
20		intex 3465	. . . 4 |- (A =/= (/) <-> |^|A e. _V)
21		elintg 3222	. . . 4 |- (|^|A e. _V -> (|^|A e. |^|(A \ {|^|A}) <-> A.x e. (A \ {|^|A})|^|A e. x))
22	20, 21	sylbi 216	. . 3 |- (A =/= (/) -> (|^|A e. |^|(A \ {|^|A}) <-> A.x e. (A \ {|^|A})|^|A e. x))
23	22	adantl 424	. 2 |- ((A C_ On /\ A =/= (/)) -> (|^|A e. |^|(A \ {|^|A}) <-> A.x e. (A \ {|^|A})|^|A e. x))
24	19, 23	mpbird 213	1 |- ((A C_ On /\ A =/= (/)) -> |^|A e. |^|(A \ {|^|A}))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  _Vcvv 2292   \ cdif 2590   C_ wss 2593  (/)c0 2875  {csn 3044  |^|cint 3214  Oncon0 3657

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661

Copyright terms: Public domain