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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 13414 | . . 3 ⊢ (𝑆 ∈ 𝐴 → 〈“𝑆”〉 = {〈0, 𝑆〉}) | |
2 | s1val 13414 | . . 3 ⊢ (𝑇 ∈ 𝐴 → 〈“𝑇”〉 = {〈0, 𝑇〉}) | |
3 | 1, 2 | eqeqan12d 2667 | . 2 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (〈“𝑆”〉 = 〈“𝑇”〉 ↔ {〈0, 𝑆〉} = {〈0, 𝑇〉})) |
4 | opex 4962 | . . 3 ⊢ 〈0, 𝑆〉 ∈ V | |
5 | sneqbg 4406 | . . 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 11426 | . . . 4 ⊢ 0 ∈ ℤ | |
8 | eqid 2651 | . . . . 5 ⊢ 0 = 0 | |
9 | opthg 4975 | . . . . . 6 ⊢ ((0 ∈ ℤ ∧ 𝑆 ∈ 𝐴) → (〈0, 𝑆〉 = 〈0, 𝑇〉 ↔ (0 = 0 ∧ 𝑆 = 𝑇))) | |
10 | 9 | baibd 968 | . . . . 5 ⊢ (((0 ∈ ℤ ∧ 𝑆 ∈ 𝐴) ∧ 0 = 0) → (〈0, 𝑆〉 = 〈0, 𝑇〉 ↔ 𝑆 = 𝑇)) |
11 | 8, 10 | mpan2 707 | . . . 4 ⊢ ((0 ∈ ℤ ∧ 𝑆 ∈ 𝐴) → (〈0, 𝑆〉 = 〈0, 𝑇〉 ↔ 𝑆 = 𝑇)) |
12 | 7, 11 | mpan 706 | . . 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 294 | 1 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (〈“𝑆”〉 = 〈“𝑇”〉 ↔ 𝑆 = 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1523 ∈ wcel 2030 Vcvv 3231 {csn 4210 〈cop 4216 0cc0 9974 ℤcz 11415 〈“cs1 13326 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-i2m1 10042 ax-1ne0 10043 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-iota 5889 df-fun 5928 df-fv 5934 df-ov 6693 df-neg 10307 df-z 11416 df-s1 13334 |
This theorem is referenced by: ccats1alpha 13436 2swrd1eqwrdeq 13500 s2eq2seq 13728 s3eq3seq 13730 2swrd2eqwrdeq 13742 efgredlemc 18204 mvhf1 31582 pfxsuff1eqwrdeq 41732 |
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