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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 | ⊢ 𝐴 = (Atoms‘𝐾) |
atpsub.s | ⊢ 𝑆 = (PSubSp‘𝐾) |
Ref | Expression |
---|---|
psubatN | ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑋) → 𝑌 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | atpsub.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
2 | atpsub.s | . . . 4 ⊢ 𝑆 = (PSubSp‘𝐾) | |
3 | 1, 2 | psubssat 36894 | . . 3 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ∈ 𝑆) → 𝑋 ⊆ 𝐴) |
4 | 3 | sseld 3969 | . 2 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ∈ 𝑆) → (𝑌 ∈ 𝑋 → 𝑌 ∈ 𝐴)) |
5 | 4 | 3impia 1113 | 1 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑋) → 𝑌 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ‘cfv 6358 Atomscatm 36403 PSubSpcpsubsp 36636 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-iota 6317 df-fun 6360 df-fv 6366 df-ov 7162 df-psubsp 36643 |
This theorem is referenced by: (None) |
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