Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
||
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 6663 | . . . 4 ⊢ (𝑥 = 𝐴 → (Singleton‘𝑥) = (Singleton‘𝐴)) | |
2 | sneq 4569 | . . . 4 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
3 | 1, 2 | eqeq12d 2835 | . . 3 ⊢ (𝑥 = 𝐴 → ((Singleton‘𝑥) = {𝑥} ↔ (Singleton‘𝐴) = {𝐴})) |
4 | eqid 2819 | . . . . 5 ⊢ {𝑥} = {𝑥} | |
5 | vex 3496 | . . . . . 6 ⊢ 𝑥 ∈ V | |
6 | snex 5322 | . . . . . 6 ⊢ {𝑥} ∈ V | |
7 | 5, 6 | brsingle 33371 | . . . . 5 ⊢ (𝑥Singleton{𝑥} ↔ {𝑥} = {𝑥}) |
8 | 4, 7 | mpbir 233 | . . . 4 ⊢ 𝑥Singleton{𝑥} |
9 | fnsingle 33373 | . . . . 5 ⊢ Singleton Fn V | |
10 | fnbrfvb 6711 | . . . . 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 233 | . . 3 ⊢ (Singleton‘𝑥) = {𝑥} |
13 | 3, 12 | vtoclg 3566 | . 2 ⊢ (𝐴 ∈ V → (Singleton‘𝐴) = {𝐴}) |
14 | fvprc 6656 | . . 3 ⊢ (¬ 𝐴 ∈ V → (Singleton‘𝐴) = ∅) | |
15 | snprc 4645 | . . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
16 | 15 | biimpi 218 | . . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
17 | 14, 16 | eqtr4d 2857 | . 2 ⊢ (¬ 𝐴 ∈ V → (Singleton‘𝐴) = {𝐴}) |
18 | 13, 17 | pm2.61i 184 | 1 ⊢ (Singleton‘𝐴) = {𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 = wceq 1531 ∈ wcel 2108 Vcvv 3493 ∅c0 4289 {csn 4559 class class class wbr 5057 Fn wfn 6343 ‘cfv 6348 Singletoncsingle 33292 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-symdif 4217 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-eprel 5458 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fo 6354 df-fv 6356 df-1st 7681 df-2nd 7682 df-txp 33308 df-singleton 33316 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |