Theorem s111 11108

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 11094	. . 3 |- ( S e. A -> <" S "> = { <. 0 , S >. } )
2		s1val 11094	. . 3 |- ( T e. A -> <" T "> = { <. 0 , T >. } )
3	1, 2	eqeqan12d 2222	. 2 |- ( ( S e. A /\ T e. A ) -> ( <" S "> = <" T "> <-> { <. 0 , S >. } = { <. 0 , T >. } ) )
4		0nn0 9330	. . . 4 |- 0 e. NN0
5		simpl 109	. . . 4 |- ( ( S e. A /\ T e. A ) -> S e. A )
6		opexg 4280	. . . 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 3810	. . 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 9403	. . . 4 |- 0 e. ZZ
11		eqid 2206	. . . . 5 |- 0 = 0
12		opthg 4290	. . . . . 6 |- ( ( 0 e. ZZ /\ S e. A ) -> ( <. 0 , S >. = <. 0 , T >. <-> ( 0 = 0 /\ S = T ) ) )
13	12	baibd 925	. . . . 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 1373   e. wcel 2177   _Vcvv 2773   {csn 3638   <.cop 3641   0cc0 7945   NN0cn0 9315   ZZcz 9392   <"cs1 11092

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-i2m1 8050  ax-rnegex 8054

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3003  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-br 4052  df-opab 4114  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-iota 5241  df-fun 5282  df-fv 5288  df-ov 5960  df-neg 8266  df-n0 9316  df-z 9393  df-s1 11093

This theorem is referenced by:  pfxsuff1eqwrdeq  11175