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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 13800 | . . 3 ⊢ (𝑆 ∈ 𝐴 → 〈“𝑆”〉 = {〈0, 𝑆〉}) | |
2 | s1val 13800 | . . 3 ⊢ (𝑇 ∈ 𝐴 → 〈“𝑇”〉 = {〈0, 𝑇〉}) | |
3 | 1, 2 | eqeqan12d 2813 | . 2 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (〈“𝑆”〉 = 〈“𝑇”〉 ↔ {〈0, 𝑆〉} = {〈0, 𝑇〉})) |
4 | opex 5255 | . . 3 ⊢ 〈0, 𝑆〉 ∈ V | |
5 | sneqbg 4687 | . . 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 11846 | . . . 4 ⊢ 0 ∈ ℤ | |
8 | eqid 2797 | . . . . 5 ⊢ 0 = 0 | |
9 | opthg 5268 | . . . . . 6 ⊢ ((0 ∈ ℤ ∧ 𝑆 ∈ 𝐴) → (〈0, 𝑆〉 = 〈0, 𝑇〉 ↔ (0 = 0 ∧ 𝑆 = 𝑇))) | |
10 | 9 | baibd 540 | . . . . 5 ⊢ (((0 ∈ ℤ ∧ 𝑆 ∈ 𝐴) ∧ 0 = 0) → (〈0, 𝑆〉 = 〈0, 𝑇〉 ↔ 𝑆 = 𝑇)) |
11 | 8, 10 | mpan2 687 | . . . 4 ⊢ ((0 ∈ ℤ ∧ 𝑆 ∈ 𝐴) → (〈0, 𝑆〉 = 〈0, 𝑇〉 ↔ 𝑆 = 𝑇)) |
12 | 7, 11 | mpan 686 | . . 3 ⊢ (𝑆 ∈ 𝐴 → (〈0, 𝑆〉 = 〈0, 𝑇〉 ↔ 𝑆 = 𝑇)) |
13 | 12 | adantr 481 | . 2 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (〈0, 𝑆〉 = 〈0, 𝑇〉 ↔ 𝑆 = 𝑇)) |
14 | 3, 6, 13 | 3bitrd 306 | 1 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (〈“𝑆”〉 = 〈“𝑇”〉 ↔ 𝑆 = 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1525 ∈ wcel 2083 Vcvv 3440 {csn 4478 〈cop 4484 0cc0 10390 ℤcz 11835 〈“cs1 13797 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pr 5228 ax-1cn 10448 ax-addrcl 10451 ax-rnegex 10461 ax-cnre 10463 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ral 3112 df-rex 3113 df-rab 3116 df-v 3442 df-sbc 3712 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-br 4969 df-opab 5031 df-id 5355 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-iota 6196 df-fun 6234 df-fv 6240 df-ov 7026 df-neg 10726 df-z 11836 df-s1 13798 |
This theorem is referenced by: ccats1alpha 13821 pfxsuff1eqwrdeq 13901 s2eq2seq 14139 s3eq3seq 14141 2swrd2eqwrdeq 14155 efgredlemc 18602 mvhf1 32416 |
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