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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 14544 | . . 3 ⊢ (𝑆 ∈ 𝐴 → 〈“𝑆”〉 = {〈0, 𝑆〉}) | |
2 | s1val 14544 | . . 3 ⊢ (𝑇 ∈ 𝐴 → 〈“𝑇”〉 = {〈0, 𝑇〉}) | |
3 | 1, 2 | eqeqan12d 2738 | . 2 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (〈“𝑆”〉 = 〈“𝑇”〉 ↔ {〈0, 𝑆〉} = {〈0, 𝑇〉})) |
4 | opex 5454 | . . 3 ⊢ 〈0, 𝑆〉 ∈ V | |
5 | sneqbg 4836 | . . 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 12565 | . . . 4 ⊢ 0 ∈ ℤ | |
8 | eqid 2724 | . . . . 5 ⊢ 0 = 0 | |
9 | opthg 5467 | . . . . . 6 ⊢ ((0 ∈ ℤ ∧ 𝑆 ∈ 𝐴) → (〈0, 𝑆〉 = 〈0, 𝑇〉 ↔ (0 = 0 ∧ 𝑆 = 𝑇))) | |
10 | 9 | baibd 539 | . . . . 5 ⊢ (((0 ∈ ℤ ∧ 𝑆 ∈ 𝐴) ∧ 0 = 0) → (〈0, 𝑆〉 = 〈0, 𝑇〉 ↔ 𝑆 = 𝑇)) |
11 | 8, 10 | mpan2 688 | . . . 4 ⊢ ((0 ∈ ℤ ∧ 𝑆 ∈ 𝐴) → (〈0, 𝑆〉 = 〈0, 𝑇〉 ↔ 𝑆 = 𝑇)) |
12 | 7, 11 | mpan 687 | . . 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 305 | 1 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → (〈“𝑆”〉 = 〈“𝑇”〉 ↔ 𝑆 = 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3466 {csn 4620 〈cop 4626 0cc0 11105 ℤcz 12554 〈“cs1 14541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 ax-1cn 11163 ax-addrcl 11166 ax-rnegex 11176 ax-cnre 11178 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-iota 6485 df-fun 6535 df-fv 6541 df-ov 7404 df-neg 11443 df-z 12555 df-s1 14542 |
This theorem is referenced by: ccats1alpha 14565 pfxsuff1eqwrdeq 14645 s2eq2seq 14884 s3eq3seq 14886 2swrd2eqwrdeq 14900 efgredlemc 19654 mvhf1 35005 |
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