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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > psubatN | Structured version Visualization version GIF version |
Description: A member of a projective subspace is an atom. (Contributed by NM, 4-Nov-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
atpsub.a | β’ π΄ = (AtomsβπΎ) |
atpsub.s | β’ π = (PSubSpβπΎ) |
Ref | Expression |
---|---|
psubatN | β’ ((πΎ β π΅ β§ π β π β§ π β π) β π β π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | atpsub.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
2 | atpsub.s | . . . 4 β’ π = (PSubSpβπΎ) | |
3 | 1, 2 | psubssat 39231 | . . 3 β’ ((πΎ β π΅ β§ π β π) β π β π΄) |
4 | 3 | sseld 3979 | . 2 β’ ((πΎ β π΅ β§ π β π) β (π β π β π β π΄)) |
5 | 4 | 3impia 1114 | 1 β’ ((πΎ β π΅ β§ π β π β§ π β π) β π β π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 βcfv 6551 Atomscatm 38739 PSubSpcpsubsp 38973 |
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 2698 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 |
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 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-iota 6503 df-fun 6553 df-fv 6559 df-ov 7427 df-psubsp 38980 |
This theorem is referenced by: (None) |
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