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Mirrors > Home > MPE Home > Th. List > Mathboxes > leat3 | Structured version Visualization version GIF version |
Description: A poset element less than or equal to an atom is either an atom or zero. (Contributed by NM, 2-Dec-2012.) |
Ref | Expression |
---|---|
leatom.b | ⊢ 𝐵 = (Base‘𝐾) |
leatom.l | ⊢ ≤ = (le‘𝐾) |
leatom.z | ⊢ 0 = (0.‘𝐾) |
leatom.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
leat3 | ⊢ (((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑋 ≤ 𝑃) → (𝑋 ∈ 𝐴 ∨ 𝑋 = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | leatom.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | leatom.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
3 | leatom.z | . . 3 ⊢ 0 = (0.‘𝐾) | |
4 | leatom.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | 1, 2, 3, 4 | leat 35871 | . 2 ⊢ (((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑋 ≤ 𝑃) → (𝑋 = 𝑃 ∨ 𝑋 = 0 )) |
6 | simpl3 1173 | . . . 4 ⊢ (((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑋 ≤ 𝑃) → 𝑃 ∈ 𝐴) | |
7 | eleq1a 2862 | . . . 4 ⊢ (𝑃 ∈ 𝐴 → (𝑋 = 𝑃 → 𝑋 ∈ 𝐴)) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ (((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑋 ≤ 𝑃) → (𝑋 = 𝑃 → 𝑋 ∈ 𝐴)) |
9 | 8 | orim1d 948 | . 2 ⊢ (((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑋 ≤ 𝑃) → ((𝑋 = 𝑃 ∨ 𝑋 = 0 ) → (𝑋 ∈ 𝐴 ∨ 𝑋 = 0 ))) |
10 | 5, 9 | mpd 15 | 1 ⊢ (((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑋 ≤ 𝑃) → (𝑋 ∈ 𝐴 ∨ 𝑋 = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∨ wo 833 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 class class class wbr 4929 ‘cfv 6188 Basecbs 16339 lecple 16428 0.cp0 17505 OPcops 35750 Atomscatm 35841 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-id 5312 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-proset 17396 df-poset 17414 df-plt 17426 df-glb 17443 df-p0 17507 df-oposet 35754 df-covers 35844 df-ats 35845 |
This theorem is referenced by: cdleme22b 36919 |
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