Theorem s111 11207

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 11193	. . 3 |- ( S e. A -> <" S "> = { <. 0 , S >. } )
2		s1val 11193	. . 3 |- ( T e. A -> <" T "> = { <. 0 , T >. } )
3	1, 2	eqeqan12d 2247	. 2 |- ( ( S e. A /\ T e. A ) -> ( <" S "> = <" T "> <-> { <. 0 , S >. } = { <. 0 , T >. } ) )
4		0nn0 9416	. . . 4 |- 0 e. NN0
5		simpl 109	. . . 4 |- ( ( S e. A /\ T e. A ) -> S e. A )
6		opexg 4320	. . . 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 3846	. . 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 9489	. . . 4 |- 0 e. ZZ
11		eqid 2231	. . . . 5 |- 0 = 0
12		opthg 4330	. . . . . 6 |- ( ( 0 e. ZZ /\ S e. A ) -> ( <. 0 , S >. = <. 0 , T >. <-> ( 0 = 0 /\ S = T ) ) )
13	12	baibd 930	. . . . 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 1397   e. wcel 2202   _Vcvv 2802   {csn 3669   <.cop 3672   0cc0 8031   NN0cn0 9401   ZZcz 9478   <"cs1 11191

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-i2m1 8136  ax-rnegex 8140

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-ov 6020  df-neg 8352  df-n0 9402  df-z 9479  df-s1 11192

This theorem is referenced by:  pfxsuff1eqwrdeq  11279