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Mirrors > Home > MPE Home > Th. List > Mathboxes > atex | Structured version Visualization version GIF version |
Description: At least one atom exists. (Contributed by NM, 15-Jul-2012.) |
Ref | Expression |
---|---|
atex.1 | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
atex | ⊢ (𝐾 ∈ HL → 𝐴 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | atex.1 | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
2 | 1 | hl2at 37154 | . . 3 ⊢ (𝐾 ∈ HL → ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 𝑝 ≠ 𝑞) |
3 | df-rex 3064 | . . . 4 ⊢ (∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 𝑝 ≠ 𝑞 ↔ ∃𝑝(𝑝 ∈ 𝐴 ∧ ∃𝑞 ∈ 𝐴 𝑝 ≠ 𝑞)) | |
4 | exsimpl 1876 | . . . 4 ⊢ (∃𝑝(𝑝 ∈ 𝐴 ∧ ∃𝑞 ∈ 𝐴 𝑝 ≠ 𝑞) → ∃𝑝 𝑝 ∈ 𝐴) | |
5 | 3, 4 | sylbi 220 | . . 3 ⊢ (∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 𝑝 ≠ 𝑞 → ∃𝑝 𝑝 ∈ 𝐴) |
6 | 2, 5 | syl 17 | . 2 ⊢ (𝐾 ∈ HL → ∃𝑝 𝑝 ∈ 𝐴) |
7 | n0 4258 | . 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑝 𝑝 ∈ 𝐴) | |
8 | 6, 7 | sylibr 237 | 1 ⊢ (𝐾 ∈ HL → 𝐴 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∃wex 1787 ∈ wcel 2110 ≠ wne 2939 ∃wrex 3059 ∅c0 4234 ‘cfv 6377 Atomscatm 37012 HLchlt 37099 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5255 ax-pr 5319 ax-un 7520 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2940 df-ral 3063 df-rex 3064 df-reu 3065 df-rab 3067 df-v 3407 df-sbc 3692 df-csb 3809 df-dif 3866 df-un 3868 df-in 3870 df-ss 3880 df-nul 4235 df-if 4437 df-pw 4512 df-sn 4539 df-pr 4541 df-op 4545 df-uni 4817 df-iun 4903 df-br 5051 df-opab 5113 df-mpt 5133 df-id 5452 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6335 df-fun 6379 df-fn 6380 df-f 6381 df-f1 6382 df-fo 6383 df-f1o 6384 df-fv 6385 df-riota 7167 df-ov 7213 df-oprab 7214 df-proset 17799 df-poset 17817 df-plt 17833 df-lub 17849 df-glb 17850 df-join 17851 df-meet 17852 df-p0 17928 df-p1 17929 df-lat 17935 df-clat 18002 df-oposet 36925 df-ol 36927 df-oml 36928 df-covers 37015 df-ats 37016 df-atl 37047 df-cvlat 37071 df-hlat 37100 |
This theorem is referenced by: llnn0 37265 lplnn0N 37296 lvoln0N 37340 |
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