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Mirrors > Home > MPE Home > Th. List > s111 | Structured version Visualization version GIF version |
Description: The singleton word function is injective. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) |
Ref | Expression |
---|---|
s111 | ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (⟨“𝑆”⟩ = ⟨“𝑇”⟩ ↔ 𝑆 = 𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | s1val 14493 | . . 3 ⊢ (𝑆 ∈ 𝐴 → ⟨“𝑆”⟩ = {⟨0, 𝑆⟩}) | |
2 | s1val 14493 | . . 3 ⊢ (𝑇 ∈ 𝐴 → ⟨“𝑇”⟩ = {⟨0, 𝑇⟩}) | |
3 | 1, 2 | eqeqan12d 2751 | . 2 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (⟨“𝑆”⟩ = ⟨“𝑇”⟩ ↔ {⟨0, 𝑆⟩} = {⟨0, 𝑇⟩})) |
4 | opex 5426 | . . 3 ⊢ ⟨0, 𝑆⟩ ∈ V | |
5 | sneqbg 4806 | . . 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 12517 | . . . 4 ⊢ 0 ∈ ℤ | |
8 | eqid 2737 | . . . . 5 ⊢ 0 = 0 | |
9 | opthg 5439 | . . . . . 6 ⊢ ((0 ∈ ℤ ∧ 𝑆 ∈ 𝐴) → (⟨0, 𝑆⟩ = ⟨0, 𝑇⟩ ↔ (0 = 0 ∧ 𝑆 = 𝑇))) | |
10 | 9 | baibd 541 | . . . . 5 ⊢ (((0 ∈ ℤ ∧ 𝑆 ∈ 𝐴) ∧ 0 = 0) → (⟨0, 𝑆⟩ = ⟨0, 𝑇⟩ ↔ 𝑆 = 𝑇)) |
11 | 8, 10 | mpan2 690 | . . . 4 ⊢ ((0 ∈ ℤ ∧ 𝑆 ∈ 𝐴) → (⟨0, 𝑆⟩ = ⟨0, 𝑇⟩ ↔ 𝑆 = 𝑇)) |
12 | 7, 11 | mpan 689 | . . 3 ⊢ (𝑆 ∈ 𝐴 → (⟨0, 𝑆⟩ = ⟨0, 𝑇⟩ ↔ 𝑆 = 𝑇)) |
13 | 12 | adantr 482 | . 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)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3448 {csn 4591 ⟨cop 4597 0cc0 11058 ℤcz 12506 ⟨“cs1 14490 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 ax-1cn 11116 ax-addrcl 11119 ax-rnegex 11129 ax-cnre 11131 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-iota 6453 df-fun 6503 df-fv 6509 df-ov 7365 df-neg 11395 df-z 12507 df-s1 14491 |
This theorem is referenced by: ccats1alpha 14514 pfxsuff1eqwrdeq 14594 s2eq2seq 14833 s3eq3seq 14835 2swrd2eqwrdeq 14849 efgredlemc 19534 mvhf1 34193 |
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