Theorem onmindif2 7012

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

Proof of Theorem onmindif2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldifsn 4317	. . . 4 |- ( x e. ( A \ { |^| A } ) <-> ( x e. A /\ x =/= |^| A ) )
2		onnmin 7003	. . . . . . . . . 10 |- ( ( A C_ On /\ x e. A ) -> -. x e. |^| A )
3	2	adantlr 751	. . . . . . . . 9 |- ( ( ( A C_ On /\ A =/= (/) ) /\ x e. A ) -> -. x e. |^| A )
4		oninton 7000	. . . . . . . . . . 11 |- ( ( A C_ On /\ A =/= (/) ) -> |^| A e. On )
5	4	adantr 481	. . . . . . . . . 10 |- ( ( ( A C_ On /\ A =/= (/) ) /\ x e. A ) -> |^| A e. On )
6		ssel2 3598	. . . . . . . . . . 11 |- ( ( A C_ On /\ x e. A ) -> x e. On )
7	6	adantlr 751	. . . . . . . . . 10 |- ( ( ( A C_ On /\ A =/= (/) ) /\ x e. A ) -> x e. On )
8		ontri1 5757	. . . . . . . . . . 11 |- ( ( |^| A e. On /\ x e. On ) -> ( |^| A C_ x <-> -. x e. |^| A ) )
9		onsseleq 5765	. . . . . . . . . . 11 |- ( ( |^| A e. On /\ x e. On ) -> ( |^| A C_ x <-> ( |^| A e. x \/ |^| A = x ) ) )
10	8, 9	bitr3d 270	. . . . . . . . . 10 |- ( ( |^| A e. On /\ x e. On ) -> ( -. x e. |^| A <-> ( |^| A e. x \/ |^| A = x ) ) )
11	5, 7, 10	syl2anc 693	. . . . . . . . 9 |- ( ( ( A C_ On /\ A =/= (/) ) /\ x e. A ) -> ( -. x e. |^| A <-> ( |^| A e. x \/ |^| A = x ) ) )
12	3, 11	mpbid 222	. . . . . . . 8 |- ( ( ( A C_ On /\ A =/= (/) ) /\ x e. A ) -> ( |^| A e. x \/ |^| A = x ) )
13	12	ord 392	. . . . . . 7 |- ( ( ( A C_ On /\ A =/= (/) ) /\ x e. A ) -> ( -. |^| A e. x -> |^| A = x ) )
14		eqcom 2629	. . . . . . 7 |- ( |^| A = x <-> x = |^| A )
15	13, 14	syl6ib 241	. . . . . 6 |- ( ( ( A C_ On /\ A =/= (/) ) /\ x e. A ) -> ( -. |^| A e. x -> x = |^| A ) )
16	15	necon1ad 2811	. . . . 5 |- ( ( ( A C_ On /\ A =/= (/) ) /\ x e. A ) -> ( x =/= |^| A -> |^| A e. x ) )
17	16	expimpd 629	. . . 4 |- ( ( A C_ On /\ A =/= (/) ) -> ( ( x e. A /\ x =/= |^| A ) -> |^| A e. x ) )
18	1, 17	syl5bi 232	. . 3 |- ( ( A C_ On /\ A =/= (/) ) -> ( x e. ( A \ { |^| A } ) -> |^| A e. x ) )
19	18	ralrimiv 2965	. 2 |- ( ( A C_ On /\ A =/= (/) ) -> A. x e. ( A \ { |^| A } ) |^| A e. x )
20		intex 4820	. . . 4 |- ( A =/= (/) <-> |^| A e. _V )
21		elintg 4483	. . . 4 |- ( |^| A e. _V -> ( |^| A e. |^| ( A \ { |^| A } ) <-> A. x e. ( A \ { |^| A } ) |^| A e. x ) )
22	20, 21	sylbi 207	. . 3 |- ( A =/= (/) -> ( |^| A e. |^| ( A \ { |^| A } ) <-> A. x e. ( A \ { |^| A } ) |^| A e. x ) )
23	22	adantl 482	. 2 |- ( ( A C_ On /\ A =/= (/) ) -> ( |^| A e. |^| ( A \ { |^| A } ) <-> A. x e. ( A \ { |^| A } ) |^| A e. x ) )
24	19, 23	mpbird 247	1 |- ( ( A C_ On /\ A =/= (/) ) -> |^| A e. |^| ( A \ { |^| A } ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   \ cdif 3571   C_ wss 3574   (/)c0 3915   {csn 4177   |^|cint 4475   Oncon0 5723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727

This theorem is referenced by: (None)