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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 12970 | . . . 4 β’ β* β V | |
2 | 1, 1 | xpex 7734 | . . 3 β’ (β* Γ β*) β V |
3 | 1 | pwex 5369 | . . 3 β’ π« β* β V |
4 | 2, 3 | xpex 7734 | . 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 13335 | . . 3 β’ π:(β* Γ β*)βΆπ« β* |
7 | fssxp 6736 | . . 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 5313 | 1 β’ π β V |
Colors of variables: wff setvar class |
Syntax hints: β§ wa 395 = wceq 1533 β wcel 2098 {crab 3424 Vcvv 3466 β wss 3941 π« cpw 4595 class class class wbr 5139 Γ cxp 5665 βΆwf 6530 β cmpo 7404 β*cxr 11246 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-fv 6542 df-oprab 7406 df-mpo 7407 df-1st 7969 df-2nd 7970 df-xr 11251 |
This theorem is referenced by: iooex 13348 isbasisrelowl 36739 relowlpssretop 36745 |
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