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Theorem s111 13432
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 ((𝑆𝐴𝑇𝐴) → (⟨“𝑆”⟩ = ⟨“𝑇”⟩ ↔ 𝑆 = 𝑇))

Proof of Theorem s111
StepHypRef Expression
1 s1val 13414 . . 3 (𝑆𝐴 → ⟨“𝑆”⟩ = {⟨0, 𝑆⟩})
2 s1val 13414 . . 3 (𝑇𝐴 → ⟨“𝑇”⟩ = {⟨0, 𝑇⟩})
31, 2eqeqan12d 2667 . 2 ((𝑆𝐴𝑇𝐴) → (⟨“𝑆”⟩ = ⟨“𝑇”⟩ ↔ {⟨0, 𝑆⟩} = {⟨0, 𝑇⟩}))
4 opex 4962 . . 3 ⟨0, 𝑆⟩ ∈ V
5 sneqbg 4406 . . 3 (⟨0, 𝑆⟩ ∈ V → ({⟨0, 𝑆⟩} = {⟨0, 𝑇⟩} ↔ ⟨0, 𝑆⟩ = ⟨0, 𝑇⟩))
64, 5mp1i 13 . 2 ((𝑆𝐴𝑇𝐴) → ({⟨0, 𝑆⟩} = {⟨0, 𝑇⟩} ↔ ⟨0, 𝑆⟩ = ⟨0, 𝑇⟩))
7 0z 11426 . . . 4 0 ∈ ℤ
8 eqid 2651 . . . . 5 0 = 0
9 opthg 4975 . . . . . 6 ((0 ∈ ℤ ∧ 𝑆𝐴) → (⟨0, 𝑆⟩ = ⟨0, 𝑇⟩ ↔ (0 = 0 ∧ 𝑆 = 𝑇)))
109baibd 968 . . . . 5 (((0 ∈ ℤ ∧ 𝑆𝐴) ∧ 0 = 0) → (⟨0, 𝑆⟩ = ⟨0, 𝑇⟩ ↔ 𝑆 = 𝑇))
118, 10mpan2 707 . . . 4 ((0 ∈ ℤ ∧ 𝑆𝐴) → (⟨0, 𝑆⟩ = ⟨0, 𝑇⟩ ↔ 𝑆 = 𝑇))
127, 11mpan 706 . . 3 (𝑆𝐴 → (⟨0, 𝑆⟩ = ⟨0, 𝑇⟩ ↔ 𝑆 = 𝑇))
1312adantr 480 . 2 ((𝑆𝐴𝑇𝐴) → (⟨0, 𝑆⟩ = ⟨0, 𝑇⟩ ↔ 𝑆 = 𝑇))
143, 6, 133bitrd 294 1 ((𝑆𝐴𝑇𝐴) → (⟨“𝑆”⟩ = ⟨“𝑇”⟩ ↔ 𝑆 = 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  {csn 4210  cop 4216  0cc0 9974  cz 11415  ⟨“cs1 13326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-i2m1 10042  ax-1ne0 10043  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-neg 10307  df-z 11416  df-s1 13334
This theorem is referenced by:  ccats1alpha  13436  2swrd1eqwrdeq  13500  s2eq2seq  13728  s3eq3seq  13730  2swrd2eqwrdeq  13742  efgredlemc  18204  mvhf1  31582  pfxsuff1eqwrdeq  41732
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