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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 33436 | . 2 ⊢ Singleton = ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V))) | |
4 | brxp 5565 | . . 3 ⊢ (𝐴(V × V)𝐵 ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
5 | 1, 2, 4 | mpbir2an 710 | . 2 ⊢ 𝐴(V × V)𝐵 |
6 | velsn 4541 | . . 3 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
7 | 1 | ideq 5687 | . . 3 ⊢ (𝑥 I 𝐴 ↔ 𝑥 = 𝐴) |
8 | 6, 7 | bitr4i 281 | . 2 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 I 𝐴) |
9 | 1, 2, 3, 5, 8 | brtxpsd3 33470 | 1 ⊢ (𝐴Singleton𝐵 ↔ 𝐵 = {𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1538 ∈ wcel 2111 Vcvv 3441 {csn 4525 class class class wbr 5030 I cid 5424 × cxp 5517 Singletoncsingle 33412 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-symdif 4169 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-eprel 5430 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fo 6330 df-fv 6332 df-1st 7671 df-2nd 7672 df-txp 33428 df-singleton 33436 |
This theorem is referenced by: elsingles 33492 fnsingle 33493 fvsingle 33494 brapply 33512 brsuccf 33515 funpartlem 33516 |
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