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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ispointN | Structured version Visualization version GIF version |
Description: The predicate "is a point". (Contributed by NM, 2-Oct-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ispoint.a | β’ π΄ = (AtomsβπΎ) |
ispoint.p | β’ π = (PointsβπΎ) |
Ref | Expression |
---|---|
ispointN | β’ (πΎ β π· β (π β π β βπ β π΄ π = {π})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ispoint.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
2 | ispoint.p | . . . 4 β’ π = (PointsβπΎ) | |
3 | 1, 2 | pointsetN 39246 | . . 3 β’ (πΎ β π· β π = {π₯ β£ βπ β π΄ π₯ = {π}}) |
4 | 3 | eleq2d 2815 | . 2 β’ (πΎ β π· β (π β π β π β {π₯ β£ βπ β π΄ π₯ = {π}})) |
5 | vsnex 5435 | . . . . 5 β’ {π} β V | |
6 | eleq1 2817 | . . . . 5 β’ (π = {π} β (π β V β {π} β V)) | |
7 | 5, 6 | mpbiri 257 | . . . 4 β’ (π = {π} β π β V) |
8 | 7 | rexlimivw 3148 | . . 3 β’ (βπ β π΄ π = {π} β π β V) |
9 | eqeq1 2732 | . . . 4 β’ (π₯ = π β (π₯ = {π} β π = {π})) | |
10 | 9 | rexbidv 3176 | . . 3 β’ (π₯ = π β (βπ β π΄ π₯ = {π} β βπ β π΄ π = {π})) |
11 | 8, 10 | elab3 3677 | . 2 β’ (π β {π₯ β£ βπ β π΄ π₯ = {π}} β βπ β π΄ π = {π}) |
12 | 4, 11 | bitrdi 286 | 1 β’ (πΎ β π· β (π β π β βπ β π΄ π = {π})) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1533 β wcel 2098 {cab 2705 βwrex 3067 Vcvv 3473 {csn 4632 βcfv 6553 Atomscatm 38767 PointscpointsN 39000 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6505 df-fun 6555 df-fv 6561 df-pointsN 39007 |
This theorem is referenced by: atpointN 39248 pointpsubN 39256 |
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