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Theorem s111 13634
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 13617 . . 3 (𝑆𝐴 → ⟨“𝑆”⟩ = {⟨0, 𝑆⟩})
2 s1val 13617 . . 3 (𝑇𝐴 → ⟨“𝑇”⟩ = {⟨0, 𝑇⟩})
31, 2eqeqan12d 2816 . 2 ((𝑆𝐴𝑇𝐴) → (⟨“𝑆”⟩ = ⟨“𝑇”⟩ ↔ {⟨0, 𝑆⟩} = {⟨0, 𝑇⟩}))
4 opex 5124 . . 3 ⟨0, 𝑆⟩ ∈ V
5 sneqbg 4561 . . 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 11676 . . . 4 0 ∈ ℤ
8 eqid 2800 . . . . 5 0 = 0
9 opthg 5137 . . . . . 6 ((0 ∈ ℤ ∧ 𝑆𝐴) → (⟨0, 𝑆⟩ = ⟨0, 𝑇⟩ ↔ (0 = 0 ∧ 𝑆 = 𝑇)))
109baibd 536 . . . . 5 (((0 ∈ ℤ ∧ 𝑆𝐴) ∧ 0 = 0) → (⟨0, 𝑆⟩ = ⟨0, 𝑇⟩ ↔ 𝑆 = 𝑇))
118, 10mpan2 683 . . . 4 ((0 ∈ ℤ ∧ 𝑆𝐴) → (⟨0, 𝑆⟩ = ⟨0, 𝑇⟩ ↔ 𝑆 = 𝑇))
127, 11mpan 682 . . 3 (𝑆𝐴 → (⟨0, 𝑆⟩ = ⟨0, 𝑇⟩ ↔ 𝑆 = 𝑇))
1312adantr 473 . 2 ((𝑆𝐴𝑇𝐴) → (⟨0, 𝑆⟩ = ⟨0, 𝑇⟩ ↔ 𝑆 = 𝑇))
143, 6, 133bitrd 297 1 ((𝑆𝐴𝑇𝐴) → (⟨“𝑆”⟩ = ⟨“𝑇”⟩ ↔ 𝑆 = 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  Vcvv 3386  {csn 4369  cop 4375  0cc0 10225  cz 11665  ⟨“cs1 13614
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pr 5098  ax-1cn 10283  ax-addrcl 10286  ax-rnegex 10296  ax-cnre 10298
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-br 4845  df-opab 4907  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-iota 6065  df-fun 6104  df-fv 6110  df-ov 6882  df-neg 10560  df-z 11666  df-s1 13615
This theorem is referenced by:  ccats1alpha  13638  2swrd1eqwrdeqOLD  13707  pfxsuff1eqwrdeq  13741  s2eq2seq  14021  s3eq3seq  14023  2swrd2eqwrdeq  14037  2swrd2eqwrdeqOLD  14038  efgredlemc  18471  mvhf1  31972
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