Theorem modom 6987

Description: Two ways to express "at most one". (Contributed by Stefan O'Rear, 28-Oct-2014.)

Assertion

Ref	Expression
modom	|- ( E* x ph <-> { x | ph } ~<_ 1o )

Proof of Theorem modom

Dummy variables u v y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1oex 6583	. . 3 |- 1o e. _V
2		0lt1o 6601	. . . . 5 |- (/) e. 1o
3	2	2a1i 27	. . . 4 |- ( E* x ph -> ( u e. { x | ph } -> (/) e. 1o ) )
4		eqidd 2230	. . . . . 6 |- ( ( E* x ph /\ ( u e. { x | ph } /\ v e. { x | ph } ) ) -> (/) = (/) )
5		df-clab 2216	. . . . . . . . 9 |- ( u e. { x | ph } <-> [ u / x ] ph )
6		df-clab 2216	. . . . . . . . 9 |- ( v e. { x | ph } <-> [ v / x ] ph )
7	5, 6	anbi12i 460	. . . . . . . 8 |- ( ( u e. { x | ph } /\ v e. { x | ph } ) <-> ( [ u / x ] ph /\ [ v / x ] ph ) )
8		nfv 1574	. . . . . . . . . . 11 |- F/ y ph
9	8	mo3 2132	. . . . . . . . . 10 |- ( E* x ph <-> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
10		nfcv 2372	. . . . . . . . . . . 12 |- F/_ x u
11		nfs1v 1990	. . . . . . . . . . . . . . 15 |- F/ x [ u / x ] ph
12		nfs1v 1990	. . . . . . . . . . . . . . 15 |- F/ x [ y / x ] ph
13	11, 12	nfan 1611	. . . . . . . . . . . . . 14 |- F/ x ( [ u / x ] ph /\ [ y / x ] ph )
14		nfv 1574	. . . . . . . . . . . . . 14 |- F/ x u = y
15	13, 14	nfim 1618	. . . . . . . . . . . . 13 |- F/ x ( ( [ u / x ] ph /\ [ y / x ] ph ) -> u = y )
16	15	nfal 1622	. . . . . . . . . . . 12 |- F/ x A. y ( ( [ u / x ] ph /\ [ y / x ] ph ) -> u = y )
17		sbequ12 1817	. . . . . . . . . . . . . . 15 |- ( x = u -> ( ph <-> [ u / x ] ph ) )
18	17	anbi1d 465	. . . . . . . . . . . . . 14 |- ( x = u -> ( ( ph /\ [ y / x ] ph ) <-> ( [ u / x ] ph /\ [ y / x ] ph ) ) )
19		equequ1 1758	. . . . . . . . . . . . . 14 |- ( x = u -> ( x = y <-> u = y ) )
20	18, 19	imbi12d 234	. . . . . . . . . . . . 13 |- ( x = u -> ( ( ( ph /\ [ y / x ] ph ) -> x = y ) <-> ( ( [ u / x ] ph /\ [ y / x ] ph ) -> u = y ) ) )
21	20	albidv 1870	. . . . . . . . . . . 12 |- ( x = u -> ( A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) <-> A. y ( ( [ u / x ] ph /\ [ y / x ] ph ) -> u = y ) ) )
22	10, 16, 21	spcgf 2886	. . . . . . . . . . 11 |- ( u e. _V -> ( A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) -> A. y ( ( [ u / x ] ph /\ [ y / x ] ph ) -> u = y ) ) )
23	22	elv 2804	. . . . . . . . . 10 |- ( A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) -> A. y ( ( [ u / x ] ph /\ [ y / x ] ph ) -> u = y ) )
24	9, 23	sylbi 121	. . . . . . . . 9 |- ( E* x ph -> A. y ( ( [ u / x ] ph /\ [ y / x ] ph ) -> u = y ) )
25		nfcv 2372	. . . . . . . . . . 11 |- F/_ y v
26		nfv 1574	. . . . . . . . . . 11 |- F/ y ( ( [ u / x ] ph /\ [ v / x ] ph ) -> u = v )
27		sbequ 1886	. . . . . . . . . . . . 13 |- ( y = v -> ( [ y / x ] ph <-> [ v / x ] ph ) )
28	27	anbi2d 464	. . . . . . . . . . . 12 |- ( y = v -> ( ( [ u / x ] ph /\ [ y / x ] ph ) <-> ( [ u / x ] ph /\ [ v / x ] ph ) ) )
29		equequ2 1759	. . . . . . . . . . . 12 |- ( y = v -> ( u = y <-> u = v ) )
30	28, 29	imbi12d 234	. . . . . . . . . . 11 |- ( y = v -> ( ( ( [ u / x ] ph /\ [ y / x ] ph ) -> u = y ) <-> ( ( [ u / x ] ph /\ [ v / x ] ph ) -> u = v ) ) )
31	25, 26, 30	spcgf 2886	. . . . . . . . . 10 |- ( v e. _V -> ( A. y ( ( [ u / x ] ph /\ [ y / x ] ph ) -> u = y ) -> ( ( [ u / x ] ph /\ [ v / x ] ph ) -> u = v ) ) )
32	31	elv 2804	. . . . . . . . 9 |- ( A. y ( ( [ u / x ] ph /\ [ y / x ] ph ) -> u = y ) -> ( ( [ u / x ] ph /\ [ v / x ] ph ) -> u = v ) )
33	24, 32	syl 14	. . . . . . . 8 |- ( E* x ph -> ( ( [ u / x ] ph /\ [ v / x ] ph ) -> u = v ) )
34	7, 33	biimtrid 152	. . . . . . 7 |- ( E* x ph -> ( ( u e. { x | ph } /\ v e. { x | ph } ) -> u = v ) )
35	34	imp 124	. . . . . 6 |- ( ( E* x ph /\ ( u e. { x | ph } /\ v e. { x | ph } ) ) -> u = v )
36	4, 35	2thd 175	. . . . 5 |- ( ( E* x ph /\ ( u e. { x | ph } /\ v e. { x | ph } ) ) -> ( (/) = (/) <-> u = v ) )
37	36	ex 115	. . . 4 |- ( E* x ph -> ( ( u e. { x | ph } /\ v e. { x | ph } ) -> ( (/) = (/) <-> u = v ) ) )
38	3, 37	dom2d 6939	. . 3 |- ( E* x ph -> ( 1o e. _V -> { x | ph } ~<_ 1o ) )
39	1, 38	mpi 15	. 2 |- ( E* x ph -> { x | ph } ~<_ 1o )
40		nfab1 2374	. . . . 5 |- F/_ x { x | ph }
41		nfcv 2372	. . . . 5 |- F/_ x ~<_
42		nfcv 2372	. . . . 5 |- F/_ x 1o
43	40, 41, 42	nfbr 4131	. . . 4 |- F/ x { x | ph } ~<_ 1o
44		simpl 109	. . . . . . 7 |- ( ( { x | ph } ~<_ 1o /\ ( ph /\ [ y / x ] ph ) ) -> { x | ph } ~<_ 1o )
45		abid 2217	. . . . . . . . 9 |- ( x e. { x | ph } <-> ph )
46	45	biimpri 133	. . . . . . . 8 |- ( ph -> x e. { x | ph } )
47	46	ad2antrl 490	. . . . . . 7 |- ( ( { x | ph } ~<_ 1o /\ ( ph /\ [ y / x ] ph ) ) -> x e. { x | ph } )
48		df-clab 2216	. . . . . . . . 9 |- ( y e. { x | ph } <-> [ y / x ] ph )
49	48	biimpri 133	. . . . . . . 8 |- ( [ y / x ] ph -> y e. { x | ph } )
50	49	ad2antll 491	. . . . . . 7 |- ( ( { x | ph } ~<_ 1o /\ ( ph /\ [ y / x ] ph ) ) -> y e. { x | ph } )
51		1dom1el 6986	. . . . . . 7 |- ( ( { x | ph } ~<_ 1o /\ x e. { x | ph } /\ y e. { x | ph } ) -> x = y )
52	44, 47, 50, 51	syl3anc 1271	. . . . . 6 |- ( ( { x | ph } ~<_ 1o /\ ( ph /\ [ y / x ] ph ) ) -> x = y )
53	52	ex 115	. . . . 5 |- ( { x | ph } ~<_ 1o -> ( ( ph /\ [ y / x ] ph ) -> x = y ) )
54	53	alrimiv 1920	. . . 4 |- ( { x | ph } ~<_ 1o -> A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
55	43, 54	alrimi 1568	. . 3 |- ( { x | ph } ~<_ 1o -> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
56	55, 9	sylibr 134	. 2 |- ( { x | ph } ~<_ 1o -> E* x ph )
57	39, 56	impbii 126	1 |- ( E* x ph <-> { x | ph } ~<_ 1o )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1393   = wceq 1395   [wsb 1808   E*wmo 2078   e. wcel 2200   {cab 2215   _Vcvv 2800   (/)c0 3492   class class class wbr 4084   1oc1o 6568   ~<_ cdom 6901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4200  ax-sep 4203  ax-nul 4211  ax-pow 4260  ax-pr 4295  ax-un 4526

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3890  df-iun 3968  df-br 4085  df-opab 4147  df-mpt 4148  df-tr 4184  df-id 4386  df-iord 4459  df-on 4461  df-suc 4464  df-xp 4727  df-rel 4728  df-cnv 4729  df-co 4730  df-dm 4731  df-rn 4732  df-res 4733  df-ima 4734  df-iota 5282  df-fun 5324  df-fn 5325  df-f 5326  df-f1 5327  df-fo 5328  df-f1o 5329  df-fv 5330  df-1o 6575  df-dom 6904

This theorem is referenced by:  modom2  6988