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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 3035 . . . 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 356 . 2 (Β¬ (𝑋 β‰  βˆ… ∧ π‘Œ β‰  βˆ…) ↔ (𝑋 = βˆ… ∨ π‘Œ = βˆ…))
4 eqid 2732 . . . 4 (leβ€˜πΎ) = (leβ€˜πΎ)
5 eqid 2732 . . . 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 4357 . . . . . . . 8 Β¬ βˆƒπ‘ž ∈ βˆ… βˆƒπ‘Ÿ ∈ π‘Œ 𝑆(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)
10 rexeq 3321 . . . . . . . 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 4357 . . . . . . . . . 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 3074 . . . . . . . 8 Β¬ βˆƒπ‘ž ∈ 𝑋 βˆƒπ‘Ÿ ∈ βˆ… 𝑆(leβ€˜πΎ)(π‘ž(joinβ€˜πΎ)π‘Ÿ)
15 rexeq 3321 . . . . . . . . 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 3178 . . . . . . . 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 855 . . . . . 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 489 . . . . 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 935 . . . . 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 868 . . . 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 510 . 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 594 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 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆƒwrex 3070   βŠ† wss 3948  βˆ…c0 4322   class class class wbr 5148  β€˜cfv 6543  (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 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  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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