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

Proof of Theorem s111

Step	Hyp	Ref	Expression
1		s1val 14544	. . 3 ⊢ (𝑆 ∈ 𝐴 → ⟨“𝑆”⟩ = {⟨0, 𝑆⟩})
2		s1val 14544	. . 3 ⊢ (𝑇 ∈ 𝐴 → ⟨“𝑇”⟩ = {⟨0, 𝑇⟩})
3	1, 2	eqeqan12d 2738	. 2 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (⟨“𝑆”⟩ = ⟨“𝑇”⟩ ↔ {⟨0, 𝑆⟩} = {⟨0, 𝑇⟩}))
4		opex 5454	. . 3 ⊢ ⟨0, 𝑆⟩ ∈ V
5		sneqbg 4836	. . 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 12565	. . . 4 ⊢ 0 ∈ ℤ
8		eqid 2724	. . . . 5 ⊢ 0 = 0
9		opthg 5467	. . . . . 6 ⊢ ((0 ∈ ℤ ∧ 𝑆 ∈ 𝐴) → (⟨0, 𝑆⟩ = ⟨0, 𝑇⟩ ↔ (0 = 0 ∧ 𝑆 = 𝑇)))
10	9	baibd 539	. . . . 5 ⊢ (((0 ∈ ℤ ∧ 𝑆 ∈ 𝐴) ∧ 0 = 0) → (⟨0, 𝑆⟩ = ⟨0, 𝑇⟩ ↔ 𝑆 = 𝑇))
11	8, 10	mpan2 688	. . . 4 ⊢ ((0 ∈ ℤ ∧ 𝑆 ∈ 𝐴) → (⟨0, 𝑆⟩ = ⟨0, 𝑇⟩ ↔ 𝑆 = 𝑇))
12	7, 11	mpan 687	. . 3 ⊢ (𝑆 ∈ 𝐴 → (⟨0, 𝑆⟩ = ⟨0, 𝑇⟩ ↔ 𝑆 = 𝑇))
13	12	adantr 480	. 2 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (⟨0, 𝑆⟩ = ⟨0, 𝑇⟩ ↔ 𝑆 = 𝑇))
14	3, 6, 13	3bitrd 305	1 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (⟨“𝑆”⟩ = ⟨“𝑇”⟩ ↔ 𝑆 = 𝑇))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  Vcvv 3466  {csn 4620  ⟨cop 4626  0cc0 11105  ℤcz 12554  ⟨“cs1 14541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-1cn 11163  ax-addrcl 11166  ax-rnegex 11176  ax-cnre 11178

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-iota 6485  df-fun 6535  df-fv 6541  df-ov 7404  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  19654  mvhf1  35005