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Mirrors > Home > MPE Home > Th. List > Mathboxes > brsingle | Structured version Visualization version GIF version |
Description: The binary relation form of the singleton function. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Ref | Expression |
---|---|
brsingle.1 | ⊢ 𝐴 ∈ V |
brsingle.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
brsingle | ⊢ (𝐴Singleton𝐵 ↔ 𝐵 = {𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brsingle.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | brsingle.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | df-singleton 34903 | . 2 ⊢ Singleton = ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V))) | |
4 | brxp 5725 | . . 3 ⊢ (𝐴(V × V)𝐵 ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
5 | 1, 2, 4 | mpbir2an 709 | . 2 ⊢ 𝐴(V × V)𝐵 |
6 | velsn 4644 | . . 3 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
7 | 1 | ideq 5852 | . . 3 ⊢ (𝑥 I 𝐴 ↔ 𝑥 = 𝐴) |
8 | 6, 7 | bitr4i 277 | . 2 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 I 𝐴) |
9 | 1, 2, 3, 5, 8 | brtxpsd3 34937 | 1 ⊢ (𝐴Singleton𝐵 ↔ 𝐵 = {𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1541 ∈ wcel 2106 Vcvv 3474 {csn 4628 class class class wbr 5148 I cid 5573 × cxp 5674 Singletoncsingle 34879 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-symdif 4242 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-eprel 5580 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fo 6549 df-fv 6551 df-1st 7977 df-2nd 7978 df-txp 34895 df-singleton 34903 |
This theorem is referenced by: elsingles 34959 fnsingle 34960 fvsingle 34961 brapply 34979 brsuccf 34982 funpartlem 34983 |
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