Theorem s111 11377

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 11363	. . 3 ⊢ (𝑆 ∈ 𝐴 → ⟨“𝑆”⟩ = {⟨0, 𝑆⟩})
2		s1val 11363	. . 3 ⊢ (𝑇 ∈ 𝐴 → ⟨“𝑇”⟩ = {⟨0, 𝑇⟩})
3	1, 2	eqeqan12d 2254	. 2 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (⟨“𝑆”⟩ = ⟨“𝑇”⟩ ↔ {⟨0, 𝑆⟩} = {⟨0, 𝑇⟩}))
4		0nn0 9557	. . . 4 ⊢ 0 ∈ ℕ0
5		simpl 109	. . . 4 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → 𝑆 ∈ 𝐴)
6		opexg 4363	. . . 4 ⊢ ((0 ∈ ℕ0 ∧ 𝑆 ∈ 𝐴) → ⟨0, 𝑆⟩ ∈ V)
7	4, 5, 6	sylancr 418	. . 3 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → ⟨0, 𝑆⟩ ∈ V)
8		sneqbg 3883	. . 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 9634	. . . 4 ⊢ 0 ∈ ℤ
11		eqid 2238	. . . . 5 ⊢ 0 = 0
12		opthg 4373	. . . . . 6 ⊢ ((0 ∈ ℤ ∧ 𝑆 ∈ 𝐴) → (⟨0, 𝑆⟩ = ⟨0, 𝑇⟩ ↔ (0 = 0 ∧ 𝑆 = 𝑇)))
13	12	baibd 935	. . . . 5 ⊢ (((0 ∈ ℤ ∧ 𝑆 ∈ 𝐴) ∧ 0 = 0) → (⟨0, 𝑆⟩ = ⟨0, 𝑇⟩ ↔ 𝑆 = 𝑇))
14	11, 13	mpan2 429	. . . 4 ⊢ ((0 ∈ ℤ ∧ 𝑆 ∈ 𝐴) → (⟨0, 𝑆⟩ = ⟨0, 𝑇⟩ ↔ 𝑆 = 𝑇))
15	10, 14	mpan 428	. . 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 1402   ∈ wcel 2209  Vcvv 2821  {csn 3705  ⟨cop 3708  0cc0 8169  ℕ₀cn0 9542  ℤcz 9623  ⟨“cs1 11361

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-i2m1 8274  ax-rnegex 8278

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-ov 6078  df-neg 8490  df-n0 9543  df-z 9624  df-s1 11362

This theorem is referenced by:  pfxsuff1eqwrdeq  11449