Theorem modom 7063

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 6657	. . 3 |- 1o e. _V
2		0lt1o 6675	. . . . 5 |- (/) e. 1o
3	2	2a1i 27	. . . 4 |- ( E* x ph -> ( u e. { x | ph } -> (/) e. 1o ) )
4		eqidd 2235	. . . . . 6 |- ( ( E* x ph /\ ( u e. { x | ph } /\ v e. { x | ph } ) ) -> (/) = (/) )
5		df-clab 2221	. . . . . . . . 9 |- ( u e. { x | ph } <-> [ u / x ] ph )
6		df-clab 2221	. . . . . . . . 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 1577	. . . . . . . . . . 11 |- F/ y ph
9	8	mo3 2137	. . . . . . . . . 10 |- ( E* x ph <-> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
10		nfcv 2386	. . . . . . . . . . . 12 |- F/_ x u
11		nfs1v 1995	. . . . . . . . . . . . . . 15 |- F/ x [ u / x ] ph
12		nfs1v 1995	. . . . . . . . . . . . . . 15 |- F/ x [ y / x ] ph
13	11, 12	nfan 1614	. . . . . . . . . . . . . 14 |- F/ x ( [ u / x ] ph /\ [ y / x ] ph )
14		nfv 1577	. . . . . . . . . . . . . 14 |- F/ x u = y
15	13, 14	nfim 1621	. . . . . . . . . . . . 13 |- F/ x ( ( [ u / x ] ph /\ [ y / x ] ph ) -> u = y )
16	15	nfal 1625	. . . . . . . . . . . 12 |- F/ x A. y ( ( [ u / x ] ph /\ [ y / x ] ph ) -> u = y )
17		sbequ12 1820	. . . . . . . . . . . . . . 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 1760	. . . . . . . . . . . . . 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 1873	. . . . . . . . . . . 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 2901	. . . . . . . . . . 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 2819	. . . . . . . . . 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 2386	. . . . . . . . . . 11 |- F/_ y v
26		nfv 1577	. . . . . . . . . . 11 |- F/ y ( ( [ u / x ] ph /\ [ v / x ] ph ) -> u = v )
27		sbequ 1889	. . . . . . . . . . . . 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 1761	. . . . . . . . . . . 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 2901	. . . . . . . . . 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 2819	. . . . . . . . 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 7014	. . 3 |- ( E* x ph -> ( 1o e. _V -> { x | ph } ~<_ 1o ) )
39	1, 38	mpi 15	. 2 |- ( E* x ph -> { x | ph } ~<_ 1o )
40		nfab1 2388	. . . . 5 |- F/_ x { x | ph }
41		nfcv 2386	. . . . 5 |- F/_ x ~<_
42		nfcv 2386	. . . . 5 |- F/_ x 1o
43	40, 41, 42	nfbr 4158	. . . 4 |- F/ x { x | ph } ~<_ 1o
44		simpl 109	. . . . . . 7 |- ( ( { x | ph } ~<_ 1o /\ ( ph /\ [ y / x ] ph ) ) -> { x | ph } ~<_ 1o )
45		abid 2222	. . . . . . . . 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 2221	. . . . . . . . 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 7062	. . . . . . 7 |- ( ( { x | ph } ~<_ 1o /\ x e. { x | ph } /\ y e. { x | ph } ) -> x = y )
52	44, 47, 50, 51	syl3anc 1274	. . . . . 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 1923	. . . 4 |- ( { x | ph } ~<_ 1o -> A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
55	43, 54	alrimi 1571	. . 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 1396   = wceq 1398   [wsb 1811   E*wmo 2083   e. wcel 2205   {cab 2220   _Vcvv 2815   (/)c0 3510   class class class wbr 4111   1oc1o 6642   ~<_ cdom 6976

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-iord 4489  df-on 4491  df-suc 4494  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-1o 6649  df-dom 6979

This theorem is referenced by:  modom2  7064