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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 6103 | . . . 4 ⊢ (𝑥 = 𝐴 → (Singleton‘𝑥) = (Singleton‘𝐴)) | |
2 | sneq 4135 | . . . 4 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
3 | 1, 2 | eqeq12d 2625 | . . 3 ⊢ (𝑥 = 𝐴 → ((Singleton‘𝑥) = {𝑥} ↔ (Singleton‘𝐴) = {𝐴})) |
4 | eqid 2610 | . . . . 5 ⊢ {𝑥} = {𝑥} | |
5 | vex 3176 | . . . . . 6 ⊢ 𝑥 ∈ V | |
6 | snex 4835 | . . . . . 6 ⊢ {𝑥} ∈ V | |
7 | 5, 6 | brsingle 31194 | . . . . 5 ⊢ (𝑥Singleton{𝑥} ↔ {𝑥} = {𝑥}) |
8 | 4, 7 | mpbir 220 | . . . 4 ⊢ 𝑥Singleton{𝑥} |
9 | fnsingle 31196 | . . . . 5 ⊢ Singleton Fn V | |
10 | fnbrfvb 6146 | . . . . 5 ⊢ ((Singleton Fn V ∧ 𝑥 ∈ V) → ((Singleton‘𝑥) = {𝑥} ↔ 𝑥Singleton{𝑥})) | |
11 | 9, 5, 10 | mp2an 704 | . . . 4 ⊢ ((Singleton‘𝑥) = {𝑥} ↔ 𝑥Singleton{𝑥}) |
12 | 8, 11 | mpbir 220 | . . 3 ⊢ (Singleton‘𝑥) = {𝑥} |
13 | 3, 12 | vtoclg 3239 | . 2 ⊢ (𝐴 ∈ V → (Singleton‘𝐴) = {𝐴}) |
14 | fvprc 6097 | . . 3 ⊢ (¬ 𝐴 ∈ V → (Singleton‘𝐴) = ∅) | |
15 | snprc 4197 | . . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
16 | 15 | biimpi 205 | . . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
17 | 14, 16 | eqtr4d 2647 | . 2 ⊢ (¬ 𝐴 ∈ V → (Singleton‘𝐴) = {𝐴}) |
18 | 13, 17 | pm2.61i 175 | 1 ⊢ (Singleton‘𝐴) = {𝐴} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 {csn 4125 class class class wbr 4583 Fn wfn 5799 ‘cfv 5804 Singletoncsingle 31114 |
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-8 1979 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-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 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-symdif 3806 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-mpt 4645 df-eprel 4949 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fo 5810 df-fv 5812 df-1st 7059 df-2nd 7060 df-txp 31130 df-singleton 31138 |
This theorem is referenced by: (None) |
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