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Mirrors > Home > MPE Home > Th. List > ixxex | Structured version Visualization version GIF version |
Description: The set of intervals of extended reals exists. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 17-Nov-2014.) |
Ref | Expression |
---|---|
ixx.1 | β’ π = (π₯ β β*, π¦ β β* β¦ {π§ β β* β£ (π₯π π§ β§ π§ππ¦)}) |
Ref | Expression |
---|---|
ixxex | β’ π β V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrex 12967 | . . . 4 β’ β* β V | |
2 | 1, 1 | xpex 7736 | . . 3 β’ (β* Γ β*) β V |
3 | 1 | pwex 5377 | . . 3 β’ π« β* β V |
4 | 2, 3 | xpex 7736 | . 2 β’ ((β* Γ β*) Γ π« β*) β V |
5 | ixx.1 | . . . 4 β’ π = (π₯ β β*, π¦ β β* β¦ {π§ β β* β£ (π₯π π§ β§ π§ππ¦)}) | |
6 | 5 | ixxf 13330 | . . 3 β’ π:(β* Γ β*)βΆπ« β* |
7 | fssxp 6742 | . . 3 β’ (π:(β* Γ β*)βΆπ« β* β π β ((β* Γ β*) Γ π« β*)) | |
8 | 6, 7 | ax-mp 5 | . 2 β’ π β ((β* Γ β*) Γ π« β*) |
9 | 4, 8 | ssexi 5321 | 1 β’ π β V |
Colors of variables: wff setvar class |
Syntax hints: β§ wa 396 = wceq 1541 β wcel 2106 {crab 3432 Vcvv 3474 β wss 3947 π« cpw 4601 class class class wbr 5147 Γ cxp 5673 βΆwf 6536 β cmpo 7407 β*cxr 11243 |
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 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 |
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-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-fv 6548 df-oprab 7409 df-mpo 7410 df-1st 7971 df-2nd 7972 df-xr 11248 |
This theorem is referenced by: iooex 13343 isbasisrelowl 36227 relowlpssretop 36233 |
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