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Theorem s111 14510
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 14493 . . 3 (𝑆𝐴 → ⟨“𝑆”⟩ = {⟨0, 𝑆⟩})
2 s1val 14493 . . 3 (𝑇𝐴 → ⟨“𝑇”⟩ = {⟨0, 𝑇⟩})
31, 2eqeqan12d 2751 . 2 ((𝑆𝐴𝑇𝐴) → (⟨“𝑆”⟩ = ⟨“𝑇”⟩ ↔ {⟨0, 𝑆⟩} = {⟨0, 𝑇⟩}))
4 opex 5426 . . 3 ⟨0, 𝑆⟩ ∈ V
5 sneqbg 4806 . . 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 12517 . . . 4 0 ∈ ℤ
8 eqid 2737 . . . . 5 0 = 0
9 opthg 5439 . . . . . 6 ((0 ∈ ℤ ∧ 𝑆𝐴) → (⟨0, 𝑆⟩ = ⟨0, 𝑇⟩ ↔ (0 = 0 ∧ 𝑆 = 𝑇)))
109baibd 541 . . . . 5 (((0 ∈ ℤ ∧ 𝑆𝐴) ∧ 0 = 0) → (⟨0, 𝑆⟩ = ⟨0, 𝑇⟩ ↔ 𝑆 = 𝑇))
118, 10mpan2 690 . . . 4 ((0 ∈ ℤ ∧ 𝑆𝐴) → (⟨0, 𝑆⟩ = ⟨0, 𝑇⟩ ↔ 𝑆 = 𝑇))
127, 11mpan 689 . . 3 (𝑆𝐴 → (⟨0, 𝑆⟩ = ⟨0, 𝑇⟩ ↔ 𝑆 = 𝑇))
1312adantr 482 . 2 ((𝑆𝐴𝑇𝐴) → (⟨0, 𝑆⟩ = ⟨0, 𝑇⟩ ↔ 𝑆 = 𝑇))
143, 6, 133bitrd 305 1 ((𝑆𝐴𝑇𝐴) → (⟨“𝑆”⟩ = ⟨“𝑇”⟩ ↔ 𝑆 = 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  Vcvv 3448  {csn 4591  cop 4597  0cc0 11058  cz 12506  ⟨“cs1 14490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-1cn 11116  ax-addrcl 11119  ax-rnegex 11129  ax-cnre 11131
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509  df-ov 7365  df-neg 11395  df-z 12507  df-s1 14491
This theorem is referenced by:  ccats1alpha  14514  pfxsuff1eqwrdeq  14594  s2eq2seq  14833  s3eq3seq  14835  2swrd2eqwrdeq  14849  efgredlemc  19534  mvhf1  34193
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