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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 33220 | . 2 ⊢ Singleton = ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V))) | |
4 | brxp 5594 | . . 3 ⊢ (𝐴(V × V)𝐵 ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
5 | 1, 2, 4 | mpbir2an 707 | . 2 ⊢ 𝐴(V × V)𝐵 |
6 | velsn 4573 | . . 3 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
7 | 1 | ideq 5716 | . . 3 ⊢ (𝑥 I 𝐴 ↔ 𝑥 = 𝐴) |
8 | 6, 7 | bitr4i 279 | . 2 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 I 𝐴) |
9 | 1, 2, 3, 5, 8 | brtxpsd3 33254 | 1 ⊢ (𝐴Singleton𝐵 ↔ 𝐵 = {𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1528 ∈ wcel 2105 Vcvv 3492 {csn 4557 class class class wbr 5057 I cid 5452 × cxp 5546 Singletoncsingle 33196 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-symdif 4216 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 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 7678 df-2nd 7679 df-txp 33212 df-singleton 33220 |
This theorem is referenced by: elsingles 33276 fnsingle 33277 fvsingle 33278 brapply 33296 brsuccf 33299 funpartlem 33300 |
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