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Theorem s111 14561
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 14544 . . 3 (𝑆𝐴 → ⟨“𝑆”⟩ = {⟨0, 𝑆⟩})
2 s1val 14544 . . 3 (𝑇𝐴 → ⟨“𝑇”⟩ = {⟨0, 𝑇⟩})
31, 2eqeqan12d 2746 . 2 ((𝑆𝐴𝑇𝐴) → (⟨“𝑆”⟩ = ⟨“𝑇”⟩ ↔ {⟨0, 𝑆⟩} = {⟨0, 𝑇⟩}))
4 opex 5463 . . 3 ⟨0, 𝑆⟩ ∈ V
5 sneqbg 4843 . . 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 12565 . . . 4 0 ∈ ℤ
8 eqid 2732 . . . . 5 0 = 0
9 opthg 5476 . . . . . 6 ((0 ∈ ℤ ∧ 𝑆𝐴) → (⟨0, 𝑆⟩ = ⟨0, 𝑇⟩ ↔ (0 = 0 ∧ 𝑆 = 𝑇)))
109baibd 540 . . . . 5 (((0 ∈ ℤ ∧ 𝑆𝐴) ∧ 0 = 0) → (⟨0, 𝑆⟩ = ⟨0, 𝑇⟩ ↔ 𝑆 = 𝑇))
118, 10mpan2 689 . . . 4 ((0 ∈ ℤ ∧ 𝑆𝐴) → (⟨0, 𝑆⟩ = ⟨0, 𝑇⟩ ↔ 𝑆 = 𝑇))
127, 11mpan 688 . . 3 (𝑆𝐴 → (⟨0, 𝑆⟩ = ⟨0, 𝑇⟩ ↔ 𝑆 = 𝑇))
1312adantr 481 . 2 ((𝑆𝐴𝑇𝐴) → (⟨0, 𝑆⟩ = ⟨0, 𝑇⟩ ↔ 𝑆 = 𝑇))
143, 6, 133bitrd 304 1 ((𝑆𝐴𝑇𝐴) → (⟨“𝑆”⟩ = ⟨“𝑇”⟩ ↔ 𝑆 = 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  Vcvv 3474  {csn 4627  cop 4633  0cc0 11106  cz 12554  ⟨“cs1 14541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-1cn 11164  ax-addrcl 11167  ax-rnegex 11177  ax-cnre 11179
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-ov 7408  df-neg 11443  df-z 12555  df-s1 14542
This theorem is referenced by:  ccats1alpha  14565  pfxsuff1eqwrdeq  14645  s2eq2seq  14884  s3eq3seq  14886  2swrd2eqwrdeq  14900  efgredlemc  19607  mvhf1  34538
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