Theorem clsk1indlem2 38340

Description: The ansatz closure function ( r e. ~P 3o |-> if ( r = { (/) } , { (/) , 1o } , r ) ) has the K2 property of expanding. (Contributed by RP, 6-Jul-2021.)

Hypothesis

Ref	Expression
clsk1indlem.k	|- K = ( r e. ~P 3o |-> if ( r = { (/) } , { (/) , 1o } , r ) )

Assertion

Ref	Expression
clsk1indlem2	|- A. s e. ~P 3o s C_ ( K ` s )

Distinct variable group:   s, r

Allowed substitution hints:   K( s, r)

Proof of Theorem clsk1indlem2

Step	Hyp	Ref	Expression
1		id 22	. . . . . . . . . 10 |- ( s = { (/) } -> s = { (/) } )
2		snsspr1 4345	. . . . . . . . . 10 |- { (/) } C_ { (/) , 1o }
3	1, 2	syl6eqss 3655	. . . . . . . . 9 |- ( s = { (/) } -> s C_ { (/) , 1o } )
4	3	ancli 574	. . . . . . . 8 |- ( s = { (/) } -> ( s = { (/) } /\ s C_ { (/) , 1o } ) )
5	4	con3i 150	. . . . . . 7 |- ( -. ( s = { (/) } /\ s C_ { (/) , 1o } ) -> -. s = { (/) } )
6		ssid 3624	. . . . . . 7 |- s C_ s
7	5, 6	jctir 561	. . . . . 6 |- ( -. ( s = { (/) } /\ s C_ { (/) , 1o } ) -> ( -. s = { (/) } /\ s C_ s ) )
8	7	orri 391	. . . . 5 |- ( ( s = { (/) } /\ s C_ { (/) , 1o } ) \/ ( -. s = { (/) } /\ s C_ s ) )
9	8	a1i 11	. . . 4 |- ( s e. ~P 3o -> ( ( s = { (/) } /\ s C_ { (/) , 1o } ) \/ ( -. s = { (/) } /\ s C_ s ) ) )
10		sseq2 3627	. . . . 5 |- ( if ( s = { (/) } , { (/) , 1o } , s ) = { (/) , 1o } -> ( s C_ if ( s = { (/) } , { (/) , 1o } , s ) <-> s C_ { (/) , 1o } ) )
11		sseq2 3627	. . . . 5 |- ( if ( s = { (/) } , { (/) , 1o } , s ) = s -> ( s C_ if ( s = { (/) } , { (/) , 1o } , s ) <-> s C_ s ) )
12	10, 11	elimif 4122	. . . 4 |- ( s C_ if ( s = { (/) } , { (/) , 1o } , s ) <-> ( ( s = { (/) } /\ s C_ { (/) , 1o } ) \/ ( -. s = { (/) } /\ s C_ s ) ) )
13	9, 12	sylibr 224	. . 3 |- ( s e. ~P 3o -> s C_ if ( s = { (/) } , { (/) , 1o } , s ) )
14		eqeq1 2626	. . . . 5 |- ( r = s -> ( r = { (/) } <-> s = { (/) } ) )
15		id 22	. . . . 5 |- ( r = s -> r = s )
16	14, 15	ifbieq2d 4111	. . . 4 |- ( r = s -> if ( r = { (/) } , { (/) , 1o } , r ) = if ( s = { (/) } , { (/) , 1o } , s ) )
17		clsk1indlem.k	. . . 4 |- K = ( r e. ~P 3o |-> if ( r = { (/) } , { (/) , 1o } , r ) )
18		prex 4909	. . . . 5 |- { (/) , 1o } e. _V
19		vex 3203	. . . . 5 |- s e. _V
20	18, 19	ifex 4156	. . . 4 |- if ( s = { (/) } , { (/) , 1o } , s ) e. _V
21	16, 17, 20	fvmpt 6282	. . 3 |- ( s e. ~P 3o -> ( K ` s ) = if ( s = { (/) } , { (/) , 1o } , s ) )
22	13, 21	sseqtr4d 3642	. 2 |- ( s e. ~P 3o -> s C_ ( K ` s ) )
23	22	rgen 2922	1 |- A. s e. ~P 3o s C_ ( K ` s )

Syntax hints:   -. wn 3   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   (/)c0 3915   ifcif 4086   ~Pcpw 4158   {csn 4177   {cpr 4179   |-> cmpt 4729   ` cfv 5888   1oc1o 7553   3oc3o 7555

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-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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-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-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896

This theorem is referenced by:  clsk1independent  38344