![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > atcvr0 | Structured version Visualization version GIF version |
Description: An atom covers zero. (atcv0 29757 analog.) (Contributed by NM, 4-Nov-2011.) |
Ref | Expression |
---|---|
atomcvr0.z | ⊢ 0 = (0.‘𝐾) |
atomcvr0.c | ⊢ 𝐶 = ( ⋖ ‘𝐾) |
atomcvr0.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
atcvr0 | ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑃 ∈ 𝐴) → 0 𝐶𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2826 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | atomcvr0.z | . . 3 ⊢ 0 = (0.‘𝐾) | |
3 | atomcvr0.c | . . 3 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
4 | atomcvr0.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | 1, 2, 3, 4 | isat 35362 | . 2 ⊢ (𝐾 ∈ 𝐷 → (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ (Base‘𝐾) ∧ 0 𝐶𝑃))) |
6 | 5 | simplbda 495 | 1 ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑃 ∈ 𝐴) → 0 𝐶𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 class class class wbr 4874 ‘cfv 6124 Basecbs 16223 0.cp0 17391 ⋖ ccvr 35338 Atomscatm 35339 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pr 5128 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-sbc 3664 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-opab 4937 df-mpt 4954 df-id 5251 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-iota 6087 df-fun 6126 df-fv 6132 df-ats 35343 |
This theorem is referenced by: 0ltat 35367 leatb 35368 atnle0 35385 atlen0 35386 atcmp 35387 atcvreq0 35390 atcvr0eq 35502 lnnat 35503 athgt 35532 ps-2 35554 lhp0lt 36079 |
Copyright terms: Public domain | W3C validator |