Theorem s111 11344

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 11330	. . 3 |- ( S e. A -> <" S "> = { <. 0 , S >. } )
2		s1val 11330	. . 3 |- ( T e. A -> <" T "> = { <. 0 , T >. } )
3	1, 2	eqeqan12d 2250	. 2 |- ( ( S e. A /\ T e. A ) -> ( <" S "> = <" T "> <-> { <. 0 , S >. } = { <. 0 , T >. } ) )
4		0nn0 9528	. . . 4 |- 0 e. NN0
5		simpl 109	. . . 4 |- ( ( S e. A /\ T e. A ) -> S e. A )
6		opexg 4349	. . . 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 3872	. . 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 9605	. . . 4 |- 0 e. ZZ
11		eqid 2234	. . . . 5 |- 0 = 0
12		opthg 4359	. . . . . 6 |- ( ( 0 e. ZZ /\ S e. A ) -> ( <. 0 , S >. = <. 0 , T >. <-> ( 0 = 0 /\ S = T ) ) )
13	12	baibd 931	. . . . 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 1398   e. wcel 2205   _Vcvv 2815   {csn 3694   <.cop 3697   0cc0 8143   NN0cn0 9513   ZZcz 9594   <"cs1 11328

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 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-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-i2m1 8248  ax-rnegex 8252

This theorem depends on definitions:  df-bi 117  df-3or 1006  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-rab 2531  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-iota 5317  df-fun 5359  df-fv 5365  df-ov 6061  df-neg 8463  df-n0 9514  df-z 9595  df-s1 11329

This theorem is referenced by:  pfxsuff1eqwrdeq  11416