Theorem s111 11159

Description: The singleton word function is injective. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 26-Feb-2016.)

Assertion

Ref	Expression
s111	|- ( ( S e. A /\ T e. A ) -> ( <" S "> = <" T "> <-> S = T ) )

Proof of Theorem s111

Step	Hyp	Ref	Expression
1		s1val 11145	. . 3 |- ( S e. A -> <" S "> = { <. 0 , S >. } )
2		s1val 11145	. . 3 |- ( T e. A -> <" T "> = { <. 0 , T >. } )
3	1, 2	eqeqan12d 2245	. 2 |- ( ( S e. A /\ T e. A ) -> ( <" S "> = <" T "> <-> { <. 0 , S >. } = { <. 0 , T >. } ) )
4		0nn0 9380	. . . 4 |- 0 e. NN0
5		simpl 109	. . . 4 |- ( ( S e. A /\ T e. A ) -> S e. A )
6		opexg 4313	. . . 4 |- ( ( 0 e. NN0 /\ S e. A ) -> <. 0 , S >. e. _V )
7	4, 5, 6	sylancr 414	. . 3 |- ( ( S e. A /\ T e. A ) -> <. 0 , S >. e. _V )
8		sneqbg 3840	. . 3 |- ( <. 0 , S >. e. _V -> ( { <. 0 , S >. } = { <. 0 , T >. } <-> <. 0 , S >. = <. 0 , T >. ) )
9	7, 8	syl 14	. 2 |- ( ( S e. A /\ T e. A ) -> ( { <. 0 , S >. } = { <. 0 , T >. } <-> <. 0 , S >. = <. 0 , T >. ) )
10		0z 9453	. . . 4 |- 0 e. ZZ
11		eqid 2229	. . . . 5 |- 0 = 0
12		opthg 4323	. . . . . 6 |- ( ( 0 e. ZZ /\ S e. A ) -> ( <. 0 , S >. = <. 0 , T >. <-> ( 0 = 0 /\ S = T ) ) )
13	12	baibd 928	. . . . 5 |- ( ( ( 0 e. ZZ /\ S e. A ) /\ 0 = 0 ) -> ( <. 0 , S >. = <. 0 , T >. <-> S = T ) )
14	11, 13	mpan2 425	. . . 4 |- ( ( 0 e. ZZ /\ S e. A ) -> ( <. 0 , S >. = <. 0 , T >. <-> S = T ) )
15	10, 14	mpan 424	. . 3 |- ( S e. A -> ( <. 0 , S >. = <. 0 , T >. <-> S = T ) )
16	15	adantr 276	. 2 |- ( ( S e. A /\ T e. A ) -> ( <. 0 , S >. = <. 0 , T >. <-> S = T ) )
17	3, 9, 16	3bitrd 214	1 |- ( ( S e. A /\ T e. A ) -> ( <" S "> = <" T "> <-> S = T ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   _Vcvv 2799   {csn 3666   <.cop 3669   0cc0 7995   NN0cn0 9365   ZZcz 9442   <"cs1 11143

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-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-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-i2m1 8100  ax-rnegex 8104

This theorem depends on definitions:  df-bi 117  df-3or 1003  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-rab 2517  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-iota 5277  df-fun 5319  df-fv 5325  df-ov 6003  df-neg 8316  df-n0 9366  df-z 9443  df-s1 11144

This theorem is referenced by:  pfxsuff1eqwrdeq  11226