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Theorem elpadd0 38983
Description: Member of projective subspace sum with at least one empty set. (Contributed by NM, 29-Dec-2011.)
Hypotheses
Ref Expression
padd0.a 𝐴 = (Atomsβ€˜πΎ)
padd0.p + = (+π‘ƒβ€˜πΎ)
Assertion
Ref Expression
elpadd0 (((𝐾 ∈ 𝐡 ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ Β¬ (𝑋 β‰  βˆ… ∧ π‘Œ β‰  βˆ…)) β†’ (𝑆 ∈ (𝑋 + π‘Œ) ↔ (𝑆 ∈ 𝑋 ∨ 𝑆 ∈ π‘Œ)))
Proof of Theorem elpadd0
Dummy variables π‘ž π‘Ÿ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 neanior 3033 . . . 4 ((𝑋 β‰  βˆ… ∧ π‘Œ β‰  βˆ…) ↔ Β¬ (𝑋 = βˆ… ∨ π‘Œ = βˆ…))
21bicomi 223 . . 3 (Β¬ (𝑋 = βˆ… ∨ π‘Œ = βˆ…) ↔ (𝑋 β‰  βˆ… ∧ π‘Œ β‰  βˆ…))
32con1bii 355 . 2 (Β¬ (𝑋 β‰  βˆ… ∧ π‘Œ β‰  βˆ…) ↔ (𝑋 = βˆ… ∨ π‘Œ = βˆ…))
4 eqid 2730 . . . 4 (leβ€˜πΎ) = (leβ€˜πΎ)
5 eqid 2730 . . . 4 (joinβ€˜πΎ) = (joinβ€˜πΎ)
6 padd0.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
7 padd0.p . . . 4 + = (+π‘ƒβ€˜πΎ)
84, 5, 6, 7elpadd 38973 . . 3 ((𝐾 ∈ 𝐡 ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) β†’ (𝑆 ∈ (𝑋 + π‘Œ) ↔ ((𝑆 ∈ 𝑋 ∨ 𝑆 ∈ π‘Œ) ∨ (𝑆 ∈ 𝐴 ∧ βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ π‘Œ 𝑆(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)))))
9 rex0 4356 . . . . . . . 8 Β¬ βˆƒπ‘ž ∈ βˆ… βˆƒπ‘Ÿ ∈ π‘Œ 𝑆(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)
10 rexeq 3319 . . . . . . . 8 (𝑋 = βˆ… β†’ (βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ π‘Œ 𝑆(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ) ↔ βˆƒπ‘ž ∈ βˆ… βˆƒπ‘Ÿ ∈ π‘Œ 𝑆(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)))
119, 10mtbiri 326 . . . . . . 7 (𝑋 = βˆ… β†’ Β¬ βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ π‘Œ 𝑆(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ))
12 rex0 4356 . . . . . . . . . 10 Β¬ βˆƒπ‘Ÿ ∈ βˆ… 𝑆(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)
1312a1i 11 . . . . . . . . 9 (π‘ž ∈ 𝑋 β†’ Β¬ βˆƒπ‘Ÿ ∈ βˆ… 𝑆(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ))
1413nrex 3072 . . . . . . . 8 Β¬ βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ βˆ… 𝑆(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)
15 rexeq 3319 . . . . . . . . 9 (π‘Œ = βˆ… β†’ (βˆƒπ‘Ÿ ∈ π‘Œ 𝑆(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ) ↔ βˆƒπ‘Ÿ ∈ βˆ… 𝑆(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)))
1615rexbidv 3176 . . . . . . . 8 (π‘Œ = βˆ… β†’ (βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ π‘Œ 𝑆(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ) ↔ βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ βˆ… 𝑆(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)))
1714, 16mtbiri 326 . . . . . . 7 (π‘Œ = βˆ… β†’ Β¬ βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ π‘Œ 𝑆(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ))
1811, 17jaoi 853 . . . . . 6 ((𝑋 = βˆ… ∨ π‘Œ = βˆ…) β†’ Β¬ βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ π‘Œ 𝑆(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ))
1918intnand 487 . . . . 5 ((𝑋 = βˆ… ∨ π‘Œ = βˆ…) β†’ Β¬ (𝑆 ∈ 𝐴 ∧ βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ π‘Œ 𝑆(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)))
20 biorf 933 . . . . 5 (Β¬ (𝑆 ∈ 𝐴 ∧ βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ π‘Œ 𝑆(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)) β†’ ((𝑆 ∈ 𝑋 ∨ 𝑆 ∈ π‘Œ) ↔ ((𝑆 ∈ 𝐴 ∧ βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ π‘Œ 𝑆(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)) ∨ (𝑆 ∈ 𝑋 ∨ 𝑆 ∈ π‘Œ))))
2119, 20syl 17 . . . 4 ((𝑋 = βˆ… ∨ π‘Œ = βˆ…) β†’ ((𝑆 ∈ 𝑋 ∨ 𝑆 ∈ π‘Œ) ↔ ((𝑆 ∈ 𝐴 ∧ βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ π‘Œ 𝑆(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)) ∨ (𝑆 ∈ 𝑋 ∨ 𝑆 ∈ π‘Œ))))
22 orcom 866 . . . 4 (((𝑆 ∈ 𝐴 ∧ βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ π‘Œ 𝑆(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)) ∨ (𝑆 ∈ 𝑋 ∨ 𝑆 ∈ π‘Œ)) ↔ ((𝑆 ∈ 𝑋 ∨ 𝑆 ∈ π‘Œ) ∨ (𝑆 ∈ 𝐴 ∧ βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ π‘Œ 𝑆(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ))))
2321, 22bitr2di 287 . . 3 ((𝑋 = βˆ… ∨ π‘Œ = βˆ…) β†’ (((𝑆 ∈ 𝑋 ∨ 𝑆 ∈ π‘Œ) ∨ (𝑆 ∈ 𝐴 ∧ βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ π‘Œ 𝑆(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ))) ↔ (𝑆 ∈ 𝑋 ∨ 𝑆 ∈ π‘Œ)))
248, 23sylan9bb 508 . 2 (((𝐾 ∈ 𝐡 ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (𝑋 = βˆ… ∨ π‘Œ = βˆ…)) β†’ (𝑆 ∈ (𝑋 + π‘Œ) ↔ (𝑆 ∈ 𝑋 ∨ 𝑆 ∈ π‘Œ)))
253, 24sylan2b 592 1 (((𝐾 ∈ 𝐡 ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ Β¬ (𝑋 β‰  βˆ… ∧ π‘Œ β‰  βˆ…)) β†’ (𝑆 ∈ (𝑋 + π‘Œ) ↔ (𝑆 ∈ 𝑋 ∨ 𝑆 ∈ π‘Œ)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  βˆƒwrex 3068   βŠ† wss 3947  βˆ…c0 4321   class class class wbr 5147  β€˜cfv 6542  (class class class)co 7411  lecple 17208  joincjn 18268  Atomscatm 38436  +𝑃cpadd 38969
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-padd 38970
This theorem is referenced by:  paddval0  38984
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