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Theorem s111 14565
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 14548 . . 3 (𝑆𝐴 → ⟨“𝑆”⟩ = {⟨0, 𝑆⟩})
2 s1val 14548 . . 3 (𝑇𝐴 → ⟨“𝑇”⟩ = {⟨0, 𝑇⟩})
31, 2eqeqan12d 2747 . 2 ((𝑆𝐴𝑇𝐴) → (⟨“𝑆”⟩ = ⟨“𝑇”⟩ ↔ {⟨0, 𝑆⟩} = {⟨0, 𝑇⟩}))
4 opex 5465 . . 3 ⟨0, 𝑆⟩ ∈ V
5 sneqbg 4845 . . 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 12569 . . . 4 0 ∈ ℤ
8 eqid 2733 . . . . 5 0 = 0
9 opthg 5478 . . . . . 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 3475  {csn 4629  cop 4635  0cc0 11110  cz 12558  ⟨“cs1 14545
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 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-1cn 11168  ax-addrcl 11171  ax-rnegex 11181  ax-cnre 11183
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-neg 11447  df-z 12559  df-s1 14546
This theorem is referenced by:  ccats1alpha  14569  pfxsuff1eqwrdeq  14649  s2eq2seq  14888  s3eq3seq  14890  2swrd2eqwrdeq  14904  efgredlemc  19613  mvhf1  34550
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