Theorem s111 13395

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 13378	. . 3 |- ( S e. A -> <" S "> = { <. 0 , S >. } )
2		s1val 13378	. . 3 |- ( T e. A -> <" T "> = { <. 0 , T >. } )
3	1, 2	eqeqan12d 2638	. 2 |- ( ( S e. A /\ T e. A ) -> ( <" S "> = <" T "> <-> { <. 0 , S >. } = { <. 0 , T >. } ) )
4		opex 4932	. . 3 |- <. 0 , S >. e. _V
5		sneqbg 4374	. . 3 |- ( <. 0 , S >. e. _V -> ( { <. 0 , S >. } = { <. 0 , T >. } <-> <. 0 , S >. = <. 0 , T >. ) )
6	4, 5	mp1i 13	. 2 |- ( ( S e. A /\ T e. A ) -> ( { <. 0 , S >. } = { <. 0 , T >. } <-> <. 0 , S >. = <. 0 , T >. ) )
7		0z 11388	. . . 4 |- 0 e. ZZ
8		eqid 2622	. . . . 5 |- 0 = 0
9		opthg 4946	. . . . . 6 |- ( ( 0 e. ZZ /\ S e. A ) -> ( <. 0 , S >. = <. 0 , T >. <-> ( 0 = 0 /\ S = T ) ) )
10	9	baibd 948	. . . . 5 |- ( ( ( 0 e. ZZ /\ S e. A ) /\ 0 = 0 ) -> ( <. 0 , S >. = <. 0 , T >. <-> S = T ) )
11	8, 10	mpan2 707	. . . 4 |- ( ( 0 e. ZZ /\ S e. A ) -> ( <. 0 , S >. = <. 0 , T >. <-> S = T ) )
12	7, 11	mpan 706	. . 3 |- ( S e. A -> ( <. 0 , S >. = <. 0 , T >. <-> S = T ) )
13	12	adantr 481	. 2 |- ( ( S e. A /\ T e. A ) -> ( <. 0 , S >. = <. 0 , T >. <-> S = T ) )
14	3, 6, 13	3bitrd 294	1 |- ( ( S e. A /\ T e. A ) -> ( <" S "> = <" T "> <-> S = T ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   <.cop 4183   0cc0 9936   ZZcz 11377   <"cs1 13294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-i2m1 10004  ax-1ne0 10005  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-neg 10269  df-z 11378  df-s1 13302

This theorem is referenced by:  2swrd1eqwrdeq  13454  s2eq2seq  13682  s3eq3seq  13684  2swrd2eqwrdeq  13696  efgredlemc  18158  mvhf1  31456  pfxsuff1eqwrdeq  41407