Theorem modom 7037

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 6633	. . 3 |- 1o e. _V
2		0lt1o 6651	. . . . 5 |- (/) e. 1o
3	2	2a1i 27	. . . 4 |- ( E* x ph -> ( u e. { x | ph } -> (/) e. 1o ) )
4		eqidd 2232	. . . . . 6 |- ( ( E* x ph /\ ( u e. { x | ph } /\ v e. { x | ph } ) ) -> (/) = (/) )
5		df-clab 2218	. . . . . . . . 9 |- ( u e. { x | ph } <-> [ u / x ] ph )
6		df-clab 2218	. . . . . . . . 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 2134	. . . . . . . . . 10 |- ( E* x ph <-> A. x A. y ( ( ph /\ [ y / x ] ph ) -> x = y ) )
10		nfcv 2375	. . . . . . . . . . . 12 |- F/_ x u
11		nfs1v 1992	. . . . . . . . . . . . . . 15 |- F/ x [ u / x ] ph
12		nfs1v 1992	. . . . . . . . . . . . . . 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 1819	. . . . . . . . . . . . . . 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 1872	. . . . . . . . . . . 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 2889	. . . . . . . . . . 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 2807	. . . . . . . . . 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 2375	. . . . . . . . . . 11 |- F/_ y v
26		nfv 1577	. . . . . . . . . . 11 |- F/ y ( ( [ u / x ] ph /\ [ v / x ] ph ) -> u = v )
27		sbequ 1888	. . . . . . . . . . . . 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 2889	. . . . . . . . . 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 2807	. . . . . . . . 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 6989	. . 3 |- ( E* x ph -> ( 1o e. _V -> { x | ph } ~<_ 1o ) )
39	1, 38	mpi 15	. 2 |- ( E* x ph -> { x | ph } ~<_ 1o )
40		nfab1 2377	. . . . 5 |- F/_ x { x | ph }
41		nfcv 2375	. . . . 5 |- F/_ x ~<_
42		nfcv 2375	. . . . 5 |- F/_ x 1o
43	40, 41, 42	nfbr 4140	. . . 4 |- F/ x { x | ph } ~<_ 1o
44		simpl 109	. . . . . . 7 |- ( ( { x | ph } ~<_ 1o /\ ( ph /\ [ y / x ] ph ) ) -> { x | ph } ~<_ 1o )
45		abid 2219	. . . . . . . . 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 2218	. . . . . . . . 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 7036	. . . . . . 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 1922	. . . 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 1810   E*wmo 2080   e. wcel 2202   {cab 2217   _Vcvv 2803   (/)c0 3496   class class class wbr 4093   1oc1o 6618   ~<_ cdom 6951

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-1o 6625  df-dom 6954

This theorem is referenced by:  modom2  7038