Theorem s111 11319

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 11305	. . 3 ⊢ (𝑆 ∈ 𝐴 → ⟨“𝑆”⟩ = {⟨0, 𝑆⟩})
2		s1val 11305	. . 3 ⊢ (𝑇 ∈ 𝐴 → ⟨“𝑇”⟩ = {⟨0, 𝑇⟩})
3	1, 2	eqeqan12d 2248	. 2 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (⟨“𝑆”⟩ = ⟨“𝑇”⟩ ↔ {⟨0, 𝑆⟩} = {⟨0, 𝑇⟩}))
4		0nn0 9511	. . . 4 ⊢ 0 ∈ ℕ0
5		simpl 109	. . . 4 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → 𝑆 ∈ 𝐴)
6		opexg 4344	. . . 4 ⊢ ((0 ∈ ℕ0 ∧ 𝑆 ∈ 𝐴) → ⟨0, 𝑆⟩ ∈ V)
7	4, 5, 6	sylancr 414	. . 3 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → ⟨0, 𝑆⟩ ∈ V)
8		sneqbg 3867	. . 3 ⊢ (⟨0, 𝑆⟩ ∈ V → ({⟨0, 𝑆⟩} = {⟨0, 𝑇⟩} ↔ ⟨0, 𝑆⟩ = ⟨0, 𝑇⟩))
9	7, 8	syl 14	. 2 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → ({⟨0, 𝑆⟩} = {⟨0, 𝑇⟩} ↔ ⟨0, 𝑆⟩ = ⟨0, 𝑇⟩))
10		0z 9588	. . . 4 ⊢ 0 ∈ ℤ
11		eqid 2232	. . . . 5 ⊢ 0 = 0
12		opthg 4354	. . . . . 6 ⊢ ((0 ∈ ℤ ∧ 𝑆 ∈ 𝐴) → (⟨0, 𝑆⟩ = ⟨0, 𝑇⟩ ↔ (0 = 0 ∧ 𝑆 = 𝑇)))
13	12	baibd 931	. . . . 5 ⊢ (((0 ∈ ℤ ∧ 𝑆 ∈ 𝐴) ∧ 0 = 0) → (⟨0, 𝑆⟩ = ⟨0, 𝑇⟩ ↔ 𝑆 = 𝑇))
14	11, 13	mpan2 425	. . . 4 ⊢ ((0 ∈ ℤ ∧ 𝑆 ∈ 𝐴) → (⟨0, 𝑆⟩ = ⟨0, 𝑇⟩ ↔ 𝑆 = 𝑇))
15	10, 14	mpan 424	. . 3 ⊢ (𝑆 ∈ 𝐴 → (⟨0, 𝑆⟩ = ⟨0, 𝑇⟩ ↔ 𝑆 = 𝑇))
16	15	adantr 276	. 2 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (⟨0, 𝑆⟩ = ⟨0, 𝑇⟩ ↔ 𝑆 = 𝑇))
17	3, 9, 16	3bitrd 214	1 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (⟨“𝑆”⟩ = ⟨“𝑇”⟩ ↔ 𝑆 = 𝑇))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2203  Vcvv 2813  {csn 3689  ⟨cop 3692  0cc0 8127  ℕ₀cn0 9496  ℤcz 9577  ⟨“cs1 11303

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-i2m1 8232  ax-rnegex 8236

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360  df-ov 6053  df-neg 8447  df-n0 9497  df-z 9578  df-s1 11304

This theorem is referenced by:  pfxsuff1eqwrdeq  11391