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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 13231 | . . 3 ⊢ (𝑆 ∈ 𝐴 → 〈“𝑆”〉 = {〈0, 𝑆〉}) | |
2 | s1val 13231 | . . 3 ⊢ (𝑇 ∈ 𝐴 → 〈“𝑇”〉 = {〈0, 𝑇〉}) | |
3 | 1, 2 | eqeqan12d 2626 | . 2 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (〈“𝑆”〉 = 〈“𝑇”〉 ↔ {〈0, 𝑆〉} = {〈0, 𝑇〉})) |
4 | opex 4859 | . . 3 ⊢ 〈0, 𝑆〉 ∈ V | |
5 | sneqbg 4314 | . . 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 11265 | . . . 4 ⊢ 0 ∈ ℤ | |
8 | eqid 2610 | . . . . 5 ⊢ 0 = 0 | |
9 | opthg 4872 | . . . . . 6 ⊢ ((0 ∈ ℤ ∧ 𝑆 ∈ 𝐴) → (〈0, 𝑆〉 = 〈0, 𝑇〉 ↔ (0 = 0 ∧ 𝑆 = 𝑇))) | |
10 | 9 | baibd 946 | . . . . 5 ⊢ (((0 ∈ ℤ ∧ 𝑆 ∈ 𝐴) ∧ 0 = 0) → (〈0, 𝑆〉 = 〈0, 𝑇〉 ↔ 𝑆 = 𝑇)) |
11 | 8, 10 | mpan2 703 | . . . 4 ⊢ ((0 ∈ ℤ ∧ 𝑆 ∈ 𝐴) → (〈0, 𝑆〉 = 〈0, 𝑇〉 ↔ 𝑆 = 𝑇)) |
12 | 7, 11 | mpan 702 | . . 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 293 | 1 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (〈“𝑆”〉 = 〈“𝑇”〉 ↔ 𝑆 = 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 {csn 4125 〈cop 4131 0cc0 9815 ℤcz 11254 〈“cs1 13149 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-i2m1 9883 ax-1ne0 9884 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-neg 10148 df-z 11255 df-s1 13157 |
This theorem is referenced by: 2swrd1eqwrdeq 13306 s2eq2seq 13532 2swrd2eqwrdeq 13544 efgredlemc 17981 mvhf1 30710 pfxsuff1eqwrdeq 40270 |
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