Theorem scott0f 32424

Description: A version of scott0 8362 with non-free variables instead of distinct variables. (Contributed by Giovanni Mascellani, 19-Aug-2018.)

Hypotheses

Ref	Expression
scott0f.1	|- F/_ y A
scott0f.2	|- F/_ x A

Assertion

Ref	Expression
scott0f	|- ( A = (/) <-> { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } = (/) )

Distinct variable group:   x, y

Allowed substitution hints:   A( x, y)

Proof of Theorem scott0f

Dummy variables z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		scott0 8362	. 2 |- ( A = (/) <-> { w e. A | A. z e. A ( rank ` w ) C_ ( rank ` z ) } = (/) )
2		scott0f.1	. . . . . . 7 |- F/_ y A
3		nfcv 2594	. . . . . . 7 |- F/_ z A
4		nfv 1763	. . . . . . 7 |- F/ z ( rank ` x ) C_ ( rank ` y )
5		nfv 1763	. . . . . . 7 |- F/ y ( rank ` x ) C_ ( rank ` z )
6		fveq2 5870	. . . . . . . 8 |- ( y = z -> ( rank ` y ) = ( rank ` z ) )
7	6	sseq2d 3462	. . . . . . 7 |- ( y = z -> ( ( rank ` x ) C_ ( rank ` y ) <-> ( rank ` x ) C_ ( rank ` z ) ) )
8	2, 3, 4, 5, 7	cbvralf 3015	. . . . . 6 |- ( A. y e. A ( rank ` x ) C_ ( rank ` y ) <-> A. z e. A ( rank ` x ) C_ ( rank ` z ) )
9	8	a1i 11	. . . . 5 |- ( x e. A -> ( A. y e. A ( rank ` x ) C_ ( rank ` y ) <-> A. z e. A ( rank ` x ) C_ ( rank ` z ) ) )
10	9	rabbiia 3035	. . . 4 |- { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } = { x e. A | A. z e. A ( rank ` x ) C_ ( rank ` z ) }
11		nfcv 2594	. . . . 5 |- F/_ w A
12		scott0f.2	. . . . 5 |- F/_ x A
13		nfv 1763	. . . . . 6 |- F/ x ( rank ` w ) C_ ( rank ` z )
14	12, 13	nfral 2776	. . . . 5 |- F/ x A. z e. A ( rank ` w ) C_ ( rank ` z )
15		nfv 1763	. . . . 5 |- F/ w A. z e. A ( rank ` x ) C_ ( rank ` z )
16		fveq2 5870	. . . . . . 7 |- ( w = x -> ( rank ` w ) = ( rank ` x ) )
17	16	sseq1d 3461	. . . . . 6 |- ( w = x -> ( ( rank ` w ) C_ ( rank ` z ) <-> ( rank ` x ) C_ ( rank ` z ) ) )
18	17	ralbidv 2829	. . . . 5 |- ( w = x -> ( A. z e. A ( rank ` w ) C_ ( rank ` z ) <-> A. z e. A ( rank ` x ) C_ ( rank ` z ) ) )
19	11, 12, 14, 15, 18	cbvrab 3045	. . . 4 |- { w e. A | A. z e. A ( rank ` w ) C_ ( rank ` z ) } = { x e. A | A. z e. A ( rank ` x ) C_ ( rank ` z ) }
20	10, 19	eqtr4i 2478	. . 3 |- { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } = { w e. A | A. z e. A ( rank ` w ) C_ ( rank ` z ) }
21	20	eqeq1i 2458	. 2 |- ( { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } = (/) <-> { w e. A | A. z e. A ( rank ` w ) C_ ( rank ` z ) } = (/) )
22	1, 21	bitr4i 256	1 |- ( A = (/) <-> { x e. A | A. y e. A ( rank ` x ) C_ ( rank ` y ) } = (/) )

Syntax hints:   <-> wb 188   = wceq 1446   e. wcel 1889   F/_wnfc 2581   A.wral 2739   {crab 2743   C_ wss 3406   (/)c0 3733   ` cfv 5585   rankcrnk 8239

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-iin 4284  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-om 6698  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-r1 8240  df-rank 8241

This theorem is referenced by:  scottn0f  32425