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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 13951 | . . 3 ⊢ (𝑆 ∈ 𝐴 → 〈“𝑆”〉 = {〈0, 𝑆〉}) | |
2 | s1val 13951 | . . 3 ⊢ (𝑇 ∈ 𝐴 → 〈“𝑇”〉 = {〈0, 𝑇〉}) | |
3 | 1, 2 | eqeqan12d 2838 | . 2 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (〈“𝑆”〉 = 〈“𝑇”〉 ↔ {〈0, 𝑆〉} = {〈0, 𝑇〉})) |
4 | opex 5355 | . . 3 ⊢ 〈0, 𝑆〉 ∈ V | |
5 | sneqbg 4773 | . . 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 11991 | . . . 4 ⊢ 0 ∈ ℤ | |
8 | eqid 2821 | . . . . 5 ⊢ 0 = 0 | |
9 | opthg 5368 | . . . . . 6 ⊢ ((0 ∈ ℤ ∧ 𝑆 ∈ 𝐴) → (〈0, 𝑆〉 = 〈0, 𝑇〉 ↔ (0 = 0 ∧ 𝑆 = 𝑇))) | |
10 | 9 | baibd 542 | . . . . 5 ⊢ (((0 ∈ ℤ ∧ 𝑆 ∈ 𝐴) ∧ 0 = 0) → (〈0, 𝑆〉 = 〈0, 𝑇〉 ↔ 𝑆 = 𝑇)) |
11 | 8, 10 | mpan2 689 | . . . 4 ⊢ ((0 ∈ ℤ ∧ 𝑆 ∈ 𝐴) → (〈0, 𝑆〉 = 〈0, 𝑇〉 ↔ 𝑆 = 𝑇)) |
12 | 7, 11 | mpan 688 | . . 3 ⊢ (𝑆 ∈ 𝐴 → (〈0, 𝑆〉 = 〈0, 𝑇〉 ↔ 𝑆 = 𝑇)) |
13 | 12 | adantr 483 | . 2 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (〈0, 𝑆〉 = 〈0, 𝑇〉 ↔ 𝑆 = 𝑇)) |
14 | 3, 6, 13 | 3bitrd 307 | 1 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (〈“𝑆”〉 = 〈“𝑇”〉 ↔ 𝑆 = 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 {csn 4566 〈cop 4572 0cc0 10536 ℤcz 11980 〈“cs1 13948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 ax-1cn 10594 ax-addrcl 10597 ax-rnegex 10607 ax-cnre 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-iota 6313 df-fun 6356 df-fv 6362 df-ov 7158 df-neg 10872 df-z 11981 df-s1 13949 |
This theorem is referenced by: ccats1alpha 13972 pfxsuff1eqwrdeq 14060 s2eq2seq 14298 s3eq3seq 14300 2swrd2eqwrdeq 14314 efgredlemc 18870 mvhf1 32806 |
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