Theorem s111 13817

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	⊢ ((𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (⟨“𝑆”⟩ = ⟨“𝑇”⟩ ↔ 𝑆 = 𝑇))

Proof of Theorem s111

Step	Hyp	Ref	Expression
1		s1val 13800	. . 3 ⊢ (𝑆 ∈ 𝐴 → ⟨“𝑆”⟩ = {⟨0, 𝑆⟩})
2		s1val 13800	. . 3 ⊢ (𝑇 ∈ 𝐴 → ⟨“𝑇”⟩ = {⟨0, 𝑇⟩})
3	1, 2	eqeqan12d 2813	. 2 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (⟨“𝑆”⟩ = ⟨“𝑇”⟩ ↔ {⟨0, 𝑆⟩} = {⟨0, 𝑇⟩}))
4		opex 5255	. . 3 ⊢ ⟨0, 𝑆⟩ ∈ V
5		sneqbg 4687	. . 3 ⊢ (⟨0, 𝑆⟩ ∈ V → ({⟨0, 𝑆⟩} = {⟨0, 𝑇⟩} ↔ ⟨0, 𝑆⟩ = ⟨0, 𝑇⟩))
6	4, 5	mp1i 13	. 2 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → ({⟨0, 𝑆⟩} = {⟨0, 𝑇⟩} ↔ ⟨0, 𝑆⟩ = ⟨0, 𝑇⟩))
7		0z 11846	. . . 4 ⊢ 0 ∈ ℤ
8		eqid 2797	. . . . 5 ⊢ 0 = 0
9		opthg 5268	. . . . . 6 ⊢ ((0 ∈ ℤ ∧ 𝑆 ∈ 𝐴) → (⟨0, 𝑆⟩ = ⟨0, 𝑇⟩ ↔ (0 = 0 ∧ 𝑆 = 𝑇)))
10	9	baibd 540	. . . . 5 ⊢ (((0 ∈ ℤ ∧ 𝑆 ∈ 𝐴) ∧ 0 = 0) → (⟨0, 𝑆⟩ = ⟨0, 𝑇⟩ ↔ 𝑆 = 𝑇))
11	8, 10	mpan2 687	. . . 4 ⊢ ((0 ∈ ℤ ∧ 𝑆 ∈ 𝐴) → (⟨0, 𝑆⟩ = ⟨0, 𝑇⟩ ↔ 𝑆 = 𝑇))
12	7, 11	mpan 686	. . 3 ⊢ (𝑆 ∈ 𝐴 → (⟨0, 𝑆⟩ = ⟨0, 𝑇⟩ ↔ 𝑆 = 𝑇))
13	12	adantr 481	. 2 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (⟨0, 𝑆⟩ = ⟨0, 𝑇⟩ ↔ 𝑆 = 𝑇))
14	3, 6, 13	3bitrd 306	1 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (⟨“𝑆”⟩ = ⟨“𝑇”⟩ ↔ 𝑆 = 𝑇))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1525   ∈ wcel 2083  Vcvv 3440  {csn 4478  ⟨cop 4484  0cc0 10390  ℤcz 11835  ⟨“cs1 13797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228  ax-1cn 10448  ax-addrcl 10451  ax-rnegex 10461  ax-cnre 10463

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-br 4969  df-opab 5031  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-iota 6196  df-fun 6234  df-fv 6240  df-ov 7026  df-neg 10726  df-z 11836  df-s1 13798

This theorem is referenced by:  ccats1alpha  13821  pfxsuff1eqwrdeq  13901  s2eq2seq  14139  s3eq3seq  14141  2swrd2eqwrdeq  14155  efgredlemc  18602  mvhf1  32416