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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 33325 | . 2 ⊢ Singleton = ((V × V) ∖ ran ((V ⊗ E ) △ ( I ⊗ V))) | |
4 | brxp 5603 | . . 3 ⊢ (𝐴(V × V)𝐵 ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
5 | 1, 2, 4 | mpbir2an 709 | . 2 ⊢ 𝐴(V × V)𝐵 |
6 | velsn 4585 | . . 3 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴) | |
7 | 1 | ideq 5725 | . . 3 ⊢ (𝑥 I 𝐴 ↔ 𝑥 = 𝐴) |
8 | 6, 7 | bitr4i 280 | . 2 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 I 𝐴) |
9 | 1, 2, 3, 5, 8 | brtxpsd3 33359 | 1 ⊢ (𝐴Singleton𝐵 ↔ 𝐵 = {𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∈ wcel 2114 Vcvv 3496 {csn 4569 class class class wbr 5068 I cid 5461 × cxp 5555 Singletoncsingle 33301 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-symdif 4221 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-eprel 5467 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fo 6363 df-fv 6365 df-1st 7691 df-2nd 7692 df-txp 33317 df-singleton 33325 |
This theorem is referenced by: elsingles 33381 fnsingle 33382 fvsingle 33383 brapply 33401 brsuccf 33404 funpartlem 33405 |
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