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Mirrors > Home > MPE Home > Th. List > Mathboxes > fvsingle | Structured version Visualization version GIF version |
Description: The value of the singleton function. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Revised by Scott Fenton, 13-Apr-2018.) |
Ref | Expression |
---|---|
fvsingle | ⊢ (Singleton‘𝐴) = {𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6804 | . . . 4 ⊢ (𝑥 = 𝐴 → (Singleton‘𝑥) = (Singleton‘𝐴)) | |
2 | sneq 4575 | . . . 4 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
3 | 1, 2 | eqeq12d 2752 | . . 3 ⊢ (𝑥 = 𝐴 → ((Singleton‘𝑥) = {𝑥} ↔ (Singleton‘𝐴) = {𝐴})) |
4 | eqid 2736 | . . . . 5 ⊢ {𝑥} = {𝑥} | |
5 | vex 3441 | . . . . . 6 ⊢ 𝑥 ∈ V | |
6 | snex 5363 | . . . . . 6 ⊢ {𝑥} ∈ V | |
7 | 5, 6 | brsingle 34268 | . . . . 5 ⊢ (𝑥Singleton{𝑥} ↔ {𝑥} = {𝑥}) |
8 | 4, 7 | mpbir 230 | . . . 4 ⊢ 𝑥Singleton{𝑥} |
9 | fnsingle 34270 | . . . . 5 ⊢ Singleton Fn V | |
10 | fnbrfvb 6854 | . . . . 5 ⊢ ((Singleton Fn V ∧ 𝑥 ∈ V) → ((Singleton‘𝑥) = {𝑥} ↔ 𝑥Singleton{𝑥})) | |
11 | 9, 5, 10 | mp2an 690 | . . . 4 ⊢ ((Singleton‘𝑥) = {𝑥} ↔ 𝑥Singleton{𝑥}) |
12 | 8, 11 | mpbir 230 | . . 3 ⊢ (Singleton‘𝑥) = {𝑥} |
13 | 3, 12 | vtoclg 3510 | . 2 ⊢ (𝐴 ∈ V → (Singleton‘𝐴) = {𝐴}) |
14 | fvprc 6796 | . . 3 ⊢ (¬ 𝐴 ∈ V → (Singleton‘𝐴) = ∅) | |
15 | snprc 4657 | . . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
16 | 15 | biimpi 215 | . . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
17 | 14, 16 | eqtr4d 2779 | . 2 ⊢ (¬ 𝐴 ∈ V → (Singleton‘𝐴) = {𝐴}) |
18 | 13, 17 | pm2.61i 182 | 1 ⊢ (Singleton‘𝐴) = {𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 = wceq 1539 ∈ wcel 2104 Vcvv 3437 ∅c0 4262 {csn 4565 class class class wbr 5081 Fn wfn 6453 ‘cfv 6458 Singletoncsingle 34189 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-sep 5232 ax-nul 5239 ax-pr 5361 ax-un 7620 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3306 df-v 3439 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-symdif 4182 df-nul 4263 df-if 4466 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-br 5082 df-opab 5144 df-mpt 5165 df-id 5500 df-eprel 5506 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-iota 6410 df-fun 6460 df-fn 6461 df-f 6462 df-fo 6464 df-fv 6466 df-1st 7863 df-2nd 7864 df-txp 34205 df-singleton 34213 |
This theorem is referenced by: (None) |
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