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Theorem atcvat4i 31381
Description: A condition implying existence of an atom with the properties shown. Lemma 3.2.20 of [PtakPulmannova] p. 68. (Contributed by NM, 2-Jul-2004.) (New usage is discouraged.)
Hypothesis
Ref Expression
atcvat3.1 š“ āˆˆ Cā„‹
Assertion
Ref Expression
atcvat4i ((šµ āˆˆ HAtoms āˆ§ š¶ āˆˆ HAtoms) ā†’ ((š“ ā‰  0ā„‹ āˆ§ šµ āŠ† (š“ āˆØā„‹ š¶)) ā†’ āˆƒš‘„ āˆˆ HAtoms (š‘„ āŠ† š“ āˆ§ šµ āŠ† (š¶ āˆØā„‹ š‘„))))
Distinct variable groups:   š‘„,š“   š‘„,šµ   š‘„,š¶

Proof of Theorem atcvat4i
StepHypRef Expression
1 atcvat3.1 . . . . . . . . 9 š“ āˆˆ Cā„‹
21hatomici 31343 . . . . . . . 8 (š“ ā‰  0ā„‹ ā†’ āˆƒš‘„ āˆˆ HAtoms š‘„ āŠ† š“)
3 atelch 31328 . . . . . . . . . . . . . . 15 (š¶ āˆˆ HAtoms ā†’ š¶ āˆˆ Cā„‹ )
4 atelch 31328 . . . . . . . . . . . . . . 15 (š‘„ āˆˆ HAtoms ā†’ š‘„ āˆˆ Cā„‹ )
5 chub1 30491 . . . . . . . . . . . . . . 15 ((š¶ āˆˆ Cā„‹ āˆ§ š‘„ āˆˆ Cā„‹ ) ā†’ š¶ āŠ† (š¶ āˆØā„‹ š‘„))
63, 4, 5syl2an 597 . . . . . . . . . . . . . 14 ((š¶ āˆˆ HAtoms āˆ§ š‘„ āˆˆ HAtoms) ā†’ š¶ āŠ† (š¶ āˆØā„‹ š‘„))
7 sseq1 3970 . . . . . . . . . . . . . 14 (šµ = š¶ ā†’ (šµ āŠ† (š¶ āˆØā„‹ š‘„) ā†” š¶ āŠ† (š¶ āˆØā„‹ š‘„)))
86, 7syl5ibr 246 . . . . . . . . . . . . 13 (šµ = š¶ ā†’ ((š¶ āˆˆ HAtoms āˆ§ š‘„ āˆˆ HAtoms) ā†’ šµ āŠ† (š¶ āˆØā„‹ š‘„)))
98expd 417 . . . . . . . . . . . 12 (šµ = š¶ ā†’ (š¶ āˆˆ HAtoms ā†’ (š‘„ āˆˆ HAtoms ā†’ šµ āŠ† (š¶ āˆØā„‹ š‘„))))
109impcom 409 . . . . . . . . . . 11 ((š¶ āˆˆ HAtoms āˆ§ šµ = š¶) ā†’ (š‘„ āˆˆ HAtoms ā†’ šµ āŠ† (š¶ āˆØā„‹ š‘„)))
1110anim2d 613 . . . . . . . . . 10 ((š¶ āˆˆ HAtoms āˆ§ šµ = š¶) ā†’ ((š‘„ āŠ† š“ āˆ§ š‘„ āˆˆ HAtoms) ā†’ (š‘„ āŠ† š“ āˆ§ šµ āŠ† (š¶ āˆØā„‹ š‘„))))
1211expcomd 418 . . . . . . . . 9 ((š¶ āˆˆ HAtoms āˆ§ šµ = š¶) ā†’ (š‘„ āˆˆ HAtoms ā†’ (š‘„ āŠ† š“ ā†’ (š‘„ āŠ† š“ āˆ§ šµ āŠ† (š¶ āˆØā„‹ š‘„)))))
1312reximdvai 3159 . . . . . . . 8 ((š¶ āˆˆ HAtoms āˆ§ šµ = š¶) ā†’ (āˆƒš‘„ āˆˆ HAtoms š‘„ āŠ† š“ ā†’ āˆƒš‘„ āˆˆ HAtoms (š‘„ āŠ† š“ āˆ§ šµ āŠ† (š¶ āˆØā„‹ š‘„))))
142, 13syl5 34 . . . . . . 7 ((š¶ āˆˆ HAtoms āˆ§ šµ = š¶) ā†’ (š“ ā‰  0ā„‹ ā†’ āˆƒš‘„ āˆˆ HAtoms (š‘„ āŠ† š“ āˆ§ šµ āŠ† (š¶ āˆØā„‹ š‘„))))
1514ex 414 . . . . . 6 (š¶ āˆˆ HAtoms ā†’ (šµ = š¶ ā†’ (š“ ā‰  0ā„‹ ā†’ āˆƒš‘„ āˆˆ HAtoms (š‘„ āŠ† š“ āˆ§ šµ āŠ† (š¶ āˆØā„‹ š‘„)))))
1615a1i 11 . . . . 5 (šµ āŠ† (š“ āˆØā„‹ š¶) ā†’ (š¶ āˆˆ HAtoms ā†’ (šµ = š¶ ā†’ (š“ ā‰  0ā„‹ ā†’ āˆƒš‘„ āˆˆ HAtoms (š‘„ āŠ† š“ āˆ§ šµ āŠ† (š¶ āˆØā„‹ š‘„))))))
1716com4l 92 . . . 4 (š¶ āˆˆ HAtoms ā†’ (šµ = š¶ ā†’ (š“ ā‰  0ā„‹ ā†’ (šµ āŠ† (š“ āˆØā„‹ š¶) ā†’ āˆƒš‘„ āˆˆ HAtoms (š‘„ āŠ† š“ āˆ§ šµ āŠ† (š¶ āˆØā„‹ š‘„))))))
1817imp4a 424 . . 3 (š¶ āˆˆ HAtoms ā†’ (šµ = š¶ ā†’ ((š“ ā‰  0ā„‹ āˆ§ šµ āŠ† (š“ āˆØā„‹ š¶)) ā†’ āˆƒš‘„ āˆˆ HAtoms (š‘„ āŠ† š“ āˆ§ šµ āŠ† (š¶ āˆØā„‹ š‘„)))))
1918adantl 483 . 2 ((šµ āˆˆ HAtoms āˆ§ š¶ āˆˆ HAtoms) ā†’ (šµ = š¶ ā†’ ((š“ ā‰  0ā„‹ āˆ§ šµ āŠ† (š“ āˆØā„‹ š¶)) ā†’ āˆƒš‘„ āˆˆ HAtoms (š‘„ āŠ† š“ āˆ§ šµ āŠ† (š¶ āˆØā„‹ š‘„)))))
20 atelch 31328 . . . . . . . 8 (šµ āˆˆ HAtoms ā†’ šµ āˆˆ Cā„‹ )
21 chlejb2 30497 . . . . . . . . . . . . . . 15 ((š¶ āˆˆ Cā„‹ āˆ§ š“ āˆˆ Cā„‹ ) ā†’ (š¶ āŠ† š“ ā†” (š“ āˆØā„‹ š¶) = š“))
221, 21mpan2 690 . . . . . . . . . . . . . 14 (š¶ āˆˆ Cā„‹ ā†’ (š¶ āŠ† š“ ā†” (š“ āˆØā„‹ š¶) = š“))
2322biimpa 478 . . . . . . . . . . . . 13 ((š¶ āˆˆ Cā„‹ āˆ§ š¶ āŠ† š“) ā†’ (š“ āˆØā„‹ š¶) = š“)
2423sseq2d 3977 . . . . . . . . . . . 12 ((š¶ āˆˆ Cā„‹ āˆ§ š¶ āŠ† š“) ā†’ (šµ āŠ† (š“ āˆØā„‹ š¶) ā†” šµ āŠ† š“))
2524biimpa 478 . . . . . . . . . . 11 (((š¶ āˆˆ Cā„‹ āˆ§ š¶ āŠ† š“) āˆ§ šµ āŠ† (š“ āˆØā„‹ š¶)) ā†’ šµ āŠ† š“)
2625expl 459 . . . . . . . . . 10 (š¶ āˆˆ Cā„‹ ā†’ ((š¶ āŠ† š“ āˆ§ šµ āŠ† (š“ āˆØā„‹ š¶)) ā†’ šµ āŠ† š“))
2726adantl 483 . . . . . . . . 9 ((šµ āˆˆ Cā„‹ āˆ§ š¶ āˆˆ Cā„‹ ) ā†’ ((š¶ āŠ† š“ āˆ§ šµ āŠ† (š“ āˆØā„‹ š¶)) ā†’ šµ āŠ† š“))
28 chub2 30492 . . . . . . . . 9 ((šµ āˆˆ Cā„‹ āˆ§ š¶ āˆˆ Cā„‹ ) ā†’ šµ āŠ† (š¶ āˆØā„‹ šµ))
2927, 28jctird 528 . . . . . . . 8 ((šµ āˆˆ Cā„‹ āˆ§ š¶ āˆˆ Cā„‹ ) ā†’ ((š¶ āŠ† š“ āˆ§ šµ āŠ† (š“ āˆØā„‹ š¶)) ā†’ (šµ āŠ† š“ āˆ§ šµ āŠ† (š¶ āˆØā„‹ šµ))))
3020, 3, 29syl2an 597 . . . . . . 7 ((šµ āˆˆ HAtoms āˆ§ š¶ āˆˆ HAtoms) ā†’ ((š¶ āŠ† š“ āˆ§ šµ āŠ† (š“ āˆØā„‹ š¶)) ā†’ (šµ āŠ† š“ āˆ§ šµ āŠ† (š¶ āˆØā„‹ šµ))))
31 simpl 484 . . . . . . 7 ((šµ āˆˆ HAtoms āˆ§ š¶ āˆˆ HAtoms) ā†’ šµ āˆˆ HAtoms)
3230, 31jctild 527 . . . . . 6 ((šµ āˆˆ HAtoms āˆ§ š¶ āˆˆ HAtoms) ā†’ ((š¶ āŠ† š“ āˆ§ šµ āŠ† (š“ āˆØā„‹ š¶)) ā†’ (šµ āˆˆ HAtoms āˆ§ (šµ āŠ† š“ āˆ§ šµ āŠ† (š¶ āˆØā„‹ šµ)))))
3332impl 457 . . . . 5 ((((šµ āˆˆ HAtoms āˆ§ š¶ āˆˆ HAtoms) āˆ§ š¶ āŠ† š“) āˆ§ šµ āŠ† (š“ āˆØā„‹ š¶)) ā†’ (šµ āˆˆ HAtoms āˆ§ (šµ āŠ† š“ āˆ§ šµ āŠ† (š¶ āˆØā„‹ šµ))))
34 sseq1 3970 . . . . . . 7 (š‘„ = šµ ā†’ (š‘„ āŠ† š“ ā†” šµ āŠ† š“))
35 oveq2 7366 . . . . . . . 8 (š‘„ = šµ ā†’ (š¶ āˆØā„‹ š‘„) = (š¶ āˆØā„‹ šµ))
3635sseq2d 3977 . . . . . . 7 (š‘„ = šµ ā†’ (šµ āŠ† (š¶ āˆØā„‹ š‘„) ā†” šµ āŠ† (š¶ āˆØā„‹ šµ)))
3734, 36anbi12d 632 . . . . . 6 (š‘„ = šµ ā†’ ((š‘„ āŠ† š“ āˆ§ šµ āŠ† (š¶ āˆØā„‹ š‘„)) ā†” (šµ āŠ† š“ āˆ§ šµ āŠ† (š¶ āˆØā„‹ šµ))))
3837rspcev 3580 . . . . 5 ((šµ āˆˆ HAtoms āˆ§ (šµ āŠ† š“ āˆ§ šµ āŠ† (š¶ āˆØā„‹ šµ))) ā†’ āˆƒš‘„ āˆˆ HAtoms (š‘„ āŠ† š“ āˆ§ šµ āŠ† (š¶ āˆØā„‹ š‘„)))
3933, 38syl 17 . . . 4 ((((šµ āˆˆ HAtoms āˆ§ š¶ āˆˆ HAtoms) āˆ§ š¶ āŠ† š“) āˆ§ šµ āŠ† (š“ āˆØā„‹ š¶)) ā†’ āˆƒš‘„ āˆˆ HAtoms (š‘„ āŠ† š“ āˆ§ šµ āŠ† (š¶ āˆØā„‹ š‘„)))
4039adantrl 715 . . 3 ((((šµ āˆˆ HAtoms āˆ§ š¶ āˆˆ HAtoms) āˆ§ š¶ āŠ† š“) āˆ§ (š“ ā‰  0ā„‹ āˆ§ šµ āŠ† (š“ āˆØā„‹ š¶))) ā†’ āˆƒš‘„ āˆˆ HAtoms (š‘„ āŠ† š“ āˆ§ šµ āŠ† (š¶ āˆØā„‹ š‘„)))
4140exp31 421 . 2 ((šµ āˆˆ HAtoms āˆ§ š¶ āˆˆ HAtoms) ā†’ (š¶ āŠ† š“ ā†’ ((š“ ā‰  0ā„‹ āˆ§ šµ āŠ† (š“ āˆØā„‹ š¶)) ā†’ āˆƒš‘„ āˆˆ HAtoms (š‘„ āŠ† š“ āˆ§ šµ āŠ† (š¶ āˆØā„‹ š‘„)))))
42 simpr 486 . . 3 ((š“ ā‰  0ā„‹ āˆ§ šµ āŠ† (š“ āˆØā„‹ š¶)) ā†’ šµ āŠ† (š“ āˆØā„‹ š¶))
43 ioran 983 . . . 4 (Ā¬ (šµ = š¶ āˆØ š¶ āŠ† š“) ā†” (Ā¬ šµ = š¶ āˆ§ Ā¬ š¶ āŠ† š“))
441atcvat3i 31380 . . . . . . 7 ((šµ āˆˆ HAtoms āˆ§ š¶ āˆˆ HAtoms) ā†’ (((Ā¬ šµ = š¶ āˆ§ Ā¬ š¶ āŠ† š“) āˆ§ šµ āŠ† (š“ āˆØā„‹ š¶)) ā†’ (š“ āˆ© (šµ āˆØā„‹ š¶)) āˆˆ HAtoms))
453ad2antlr 726 . . . . . . . . . . 11 (((šµ āˆˆ HAtoms āˆ§ š¶ āˆˆ HAtoms) āˆ§ ((Ā¬ šµ = š¶ āˆ§ Ā¬ š¶ āŠ† š“) āˆ§ šµ āŠ† (š“ āˆØā„‹ š¶))) ā†’ š¶ āˆˆ Cā„‹ )
4644imp 408 . . . . . . . . . . 11 (((šµ āˆˆ HAtoms āˆ§ š¶ āˆˆ HAtoms) āˆ§ ((Ā¬ šµ = š¶ āˆ§ Ā¬ š¶ āŠ† š“) āˆ§ šµ āŠ† (š“ āˆØā„‹ š¶))) ā†’ (š“ āˆ© (šµ āˆØā„‹ š¶)) āˆˆ HAtoms)
47 simpll 766 . . . . . . . . . . 11 (((šµ āˆˆ HAtoms āˆ§ š¶ āˆˆ HAtoms) āˆ§ ((Ā¬ šµ = š¶ āˆ§ Ā¬ š¶ āŠ† š“) āˆ§ šµ āŠ† (š“ āˆØā„‹ š¶))) ā†’ šµ āˆˆ HAtoms)
4845, 46, 473jca 1129 . . . . . . . . . 10 (((šµ āˆˆ HAtoms āˆ§ š¶ āˆˆ HAtoms) āˆ§ ((Ā¬ šµ = š¶ āˆ§ Ā¬ š¶ āŠ† š“) āˆ§ šµ āŠ† (š“ āˆØā„‹ š¶))) ā†’ (š¶ āˆˆ Cā„‹ āˆ§ (š“ āˆ© (šµ āˆØā„‹ š¶)) āˆˆ HAtoms āˆ§ šµ āˆˆ HAtoms))
49 inss2 4190 . . . . . . . . . . . . 13 (š“ āˆ© (šµ āˆØā„‹ š¶)) āŠ† (šµ āˆØā„‹ š¶)
50 chjcom 30490 . . . . . . . . . . . . . 14 ((šµ āˆˆ Cā„‹ āˆ§ š¶ āˆˆ Cā„‹ ) ā†’ (šµ āˆØā„‹ š¶) = (š¶ āˆØā„‹ šµ))
5120, 3, 50syl2an 597 . . . . . . . . . . . . 13 ((šµ āˆˆ HAtoms āˆ§ š¶ āˆˆ HAtoms) ā†’ (šµ āˆØā„‹ š¶) = (š¶ āˆØā„‹ šµ))
5249, 51sseqtrid 3997 . . . . . . . . . . . 12 ((šµ āˆˆ HAtoms āˆ§ š¶ āˆˆ HAtoms) ā†’ (š“ āˆ© (šµ āˆØā„‹ š¶)) āŠ† (š¶ āˆØā„‹ šµ))
5352adantr 482 . . . . . . . . . . 11 (((šµ āˆˆ HAtoms āˆ§ š¶ āˆˆ HAtoms) āˆ§ ((Ā¬ šµ = š¶ āˆ§ Ā¬ š¶ āŠ† š“) āˆ§ šµ āŠ† (š“ āˆØā„‹ š¶))) ā†’ (š“ āˆ© (šµ āˆØā„‹ š¶)) āŠ† (š¶ āˆØā„‹ šµ))
54 atnssm0 31360 . . . . . . . . . . . . . . . . 17 ((š“ āˆˆ Cā„‹ āˆ§ š¶ āˆˆ HAtoms) ā†’ (Ā¬ š¶ āŠ† š“ ā†” (š“ āˆ© š¶) = 0ā„‹))
551, 54mpan 689 . . . . . . . . . . . . . . . 16 (š¶ āˆˆ HAtoms ā†’ (Ā¬ š¶ āŠ† š“ ā†” (š“ āˆ© š¶) = 0ā„‹))
5655adantl 483 . . . . . . . . . . . . . . 15 ((šµ āˆˆ HAtoms āˆ§ š¶ āˆˆ HAtoms) ā†’ (Ā¬ š¶ āŠ† š“ ā†” (š“ āˆ© š¶) = 0ā„‹))
57 inss1 4189 . . . . . . . . . . . . . . . . . . 19 (š“ āˆ© (šµ āˆØā„‹ š¶)) āŠ† š“
58 sslin 4195 . . . . . . . . . . . . . . . . . . 19 ((š“ āˆ© (šµ āˆØā„‹ š¶)) āŠ† š“ ā†’ (š¶ āˆ© (š“ āˆ© (šµ āˆØā„‹ š¶))) āŠ† (š¶ āˆ© š“))
5957, 58ax-mp 5 . . . . . . . . . . . . . . . . . 18 (š¶ āˆ© (š“ āˆ© (šµ āˆØā„‹ š¶))) āŠ† (š¶ āˆ© š“)
60 incom 4162 . . . . . . . . . . . . . . . . . 18 (š¶ āˆ© š“) = (š“ āˆ© š¶)
6159, 60sseqtri 3981 . . . . . . . . . . . . . . . . 17 (š¶ āˆ© (š“ āˆ© (šµ āˆØā„‹ š¶))) āŠ† (š“ āˆ© š¶)
62 sseq2 3971 . . . . . . . . . . . . . . . . 17 ((š“ āˆ© š¶) = 0ā„‹ ā†’ ((š¶ āˆ© (š“ āˆ© (šµ āˆØā„‹ š¶))) āŠ† (š“ āˆ© š¶) ā†” (š¶ āˆ© (š“ āˆ© (šµ āˆØā„‹ š¶))) āŠ† 0ā„‹))
6361, 62mpbii 232 . . . . . . . . . . . . . . . 16 ((š“ āˆ© š¶) = 0ā„‹ ā†’ (š¶ āˆ© (š“ āˆ© (šµ āˆØā„‹ š¶))) āŠ† 0ā„‹)
64 simpr 486 . . . . . . . . . . . . . . . . . . 19 ((šµ āˆˆ Cā„‹ āˆ§ š¶ āˆˆ Cā„‹ ) ā†’ š¶ āˆˆ Cā„‹ )
65 chjcl 30341 . . . . . . . . . . . . . . . . . . . 20 ((šµ āˆˆ Cā„‹ āˆ§ š¶ āˆˆ Cā„‹ ) ā†’ (šµ āˆØā„‹ š¶) āˆˆ Cā„‹ )
66 chincl 30483 . . . . . . . . . . . . . . . . . . . 20 ((š“ āˆˆ Cā„‹ āˆ§ (šµ āˆØā„‹ š¶) āˆˆ Cā„‹ ) ā†’ (š“ āˆ© (šµ āˆØā„‹ š¶)) āˆˆ Cā„‹ )
671, 65, 66sylancr 588 . . . . . . . . . . . . . . . . . . 19 ((šµ āˆˆ Cā„‹ āˆ§ š¶ āˆˆ Cā„‹ ) ā†’ (š“ āˆ© (šµ āˆØā„‹ š¶)) āˆˆ Cā„‹ )
68 chincl 30483 . . . . . . . . . . . . . . . . . . 19 ((š¶ āˆˆ Cā„‹ āˆ§ (š“ āˆ© (šµ āˆØā„‹ š¶)) āˆˆ Cā„‹ ) ā†’ (š¶ āˆ© (š“ āˆ© (šµ āˆØā„‹ š¶))) āˆˆ Cā„‹ )
6964, 67, 68syl2anc 585 . . . . . . . . . . . . . . . . . 18 ((šµ āˆˆ Cā„‹ āˆ§ š¶ āˆˆ Cā„‹ ) ā†’ (š¶ āˆ© (š“ āˆ© (šµ āˆØā„‹ š¶))) āˆˆ Cā„‹ )
7020, 3, 69syl2an 597 . . . . . . . . . . . . . . . . 17 ((šµ āˆˆ HAtoms āˆ§ š¶ āˆˆ HAtoms) ā†’ (š¶ āˆ© (š“ āˆ© (šµ āˆØā„‹ š¶))) āˆˆ Cā„‹ )
71 chle0 30427 . . . . . . . . . . . . . . . . 17 ((š¶ āˆ© (š“ āˆ© (šµ āˆØā„‹ š¶))) āˆˆ Cā„‹ ā†’ ((š¶ āˆ© (š“ āˆ© (šµ āˆØā„‹ š¶))) āŠ† 0ā„‹ ā†” (š¶ āˆ© (š“ āˆ© (šµ āˆØā„‹ š¶))) = 0ā„‹))
7270, 71syl 17 . . . . . . . . . . . . . . . 16 ((šµ āˆˆ HAtoms āˆ§ š¶ āˆˆ HAtoms) ā†’ ((š¶ āˆ© (š“ āˆ© (šµ āˆØā„‹ š¶))) āŠ† 0ā„‹ ā†” (š¶ āˆ© (š“ āˆ© (šµ āˆØā„‹ š¶))) = 0ā„‹))
7363, 72imbitrid 243 . . . . . . . . . . . . . . 15 ((šµ āˆˆ HAtoms āˆ§ š¶ āˆˆ HAtoms) ā†’ ((š“ āˆ© š¶) = 0ā„‹ ā†’ (š¶ āˆ© (š“ āˆ© (šµ āˆØā„‹ š¶))) = 0ā„‹))
7456, 73sylbid 239 . . . . . . . . . . . . . 14 ((šµ āˆˆ HAtoms āˆ§ š¶ āˆˆ HAtoms) ā†’ (Ā¬ š¶ āŠ† š“ ā†’ (š¶ āˆ© (š“ āˆ© (šµ āˆØā„‹ š¶))) = 0ā„‹))
7574imp 408 . . . . . . . . . . . . 13 (((šµ āˆˆ HAtoms āˆ§ š¶ āˆˆ HAtoms) āˆ§ Ā¬ š¶ āŠ† š“) ā†’ (š¶ āˆ© (š“ āˆ© (šµ āˆØā„‹ š¶))) = 0ā„‹)
7675adantrl 715 . . . . . . . . . . . 12 (((šµ āˆˆ HAtoms āˆ§ š¶ āˆˆ HAtoms) āˆ§ (Ā¬ šµ = š¶ āˆ§ Ā¬ š¶ āŠ† š“)) ā†’ (š¶ āˆ© (š“ āˆ© (šµ āˆØā„‹ š¶))) = 0ā„‹)
7776adantrr 716 . . . . . . . . . . 11 (((šµ āˆˆ HAtoms āˆ§ š¶ āˆˆ HAtoms) āˆ§ ((Ā¬ šµ = š¶ āˆ§ Ā¬ š¶ āŠ† š“) āˆ§ šµ āŠ† (š“ āˆØā„‹ š¶))) ā†’ (š¶ āˆ© (š“ āˆ© (šµ āˆØā„‹ š¶))) = 0ā„‹)
7853, 77jca 513 . . . . . . . . . 10 (((šµ āˆˆ HAtoms āˆ§ š¶ āˆˆ HAtoms) āˆ§ ((Ā¬ šµ = š¶ āˆ§ Ā¬ š¶ āŠ† š“) āˆ§ šµ āŠ† (š“ āˆØā„‹ š¶))) ā†’ ((š“ āˆ© (šµ āˆØā„‹ š¶)) āŠ† (š¶ āˆØā„‹ šµ) āˆ§ (š¶ āˆ© (š“ āˆ© (šµ āˆØā„‹ š¶))) = 0ā„‹))
79 atexch 31365 . . . . . . . . . 10 ((š¶ āˆˆ Cā„‹ āˆ§ (š“ āˆ© (šµ āˆØā„‹ š¶)) āˆˆ HAtoms āˆ§ šµ āˆˆ HAtoms) ā†’ (((š“ āˆ© (šµ āˆØā„‹ š¶)) āŠ† (š¶ āˆØā„‹ šµ) āˆ§ (š¶ āˆ© (š“ āˆ© (šµ āˆØā„‹ š¶))) = 0ā„‹) ā†’ šµ āŠ† (š¶ āˆØā„‹ (š“ āˆ© (šµ āˆØā„‹ š¶)))))
8048, 78, 79sylc 65 . . . . . . . . 9 (((šµ āˆˆ HAtoms āˆ§ š¶ āˆˆ HAtoms) āˆ§ ((Ā¬ šµ = š¶ āˆ§ Ā¬ š¶ āŠ† š“) āˆ§ šµ āŠ† (š“ āˆØā„‹ š¶))) ā†’ šµ āŠ† (š¶ āˆØā„‹ (š“ āˆ© (šµ āˆØā„‹ š¶))))
8180, 57jctil 521 . . . . . . . 8 (((šµ āˆˆ HAtoms āˆ§ š¶ āˆˆ HAtoms) āˆ§ ((Ā¬ šµ = š¶ āˆ§ Ā¬ š¶ āŠ† š“) āˆ§ šµ āŠ† (š“ āˆØā„‹ š¶))) ā†’ ((š“ āˆ© (šµ āˆØā„‹ š¶)) āŠ† š“ āˆ§ šµ āŠ† (š¶ āˆØā„‹ (š“ āˆ© (šµ āˆØā„‹ š¶)))))
8281ex 414 . . . . . . 7 ((šµ āˆˆ HAtoms āˆ§ š¶ āˆˆ HAtoms) ā†’ (((Ā¬ šµ = š¶ āˆ§ Ā¬ š¶ āŠ† š“) āˆ§ šµ āŠ† (š“ āˆØā„‹ š¶)) ā†’ ((š“ āˆ© (šµ āˆØā„‹ š¶)) āŠ† š“ āˆ§ šµ āŠ† (š¶ āˆØā„‹ (š“ āˆ© (šµ āˆØā„‹ š¶))))))
8344, 82jcad 514 . . . . . 6 ((šµ āˆˆ HAtoms āˆ§ š¶ āˆˆ HAtoms) ā†’ (((Ā¬ šµ = š¶ āˆ§ Ā¬ š¶ āŠ† š“) āˆ§ šµ āŠ† (š“ āˆØā„‹ š¶)) ā†’ ((š“ āˆ© (šµ āˆØā„‹ š¶)) āˆˆ HAtoms āˆ§ ((š“ āˆ© (šµ āˆØā„‹ š¶)) āŠ† š“ āˆ§ šµ āŠ† (š¶ āˆØā„‹ (š“ āˆ© (šµ āˆØā„‹ š¶)))))))
84 sseq1 3970 . . . . . . . 8 (š‘„ = (š“ āˆ© (šµ āˆØā„‹ š¶)) ā†’ (š‘„ āŠ† š“ ā†” (š“ āˆ© (šµ āˆØā„‹ š¶)) āŠ† š“))
85 oveq2 7366 . . . . . . . . 9 (š‘„ = (š“ āˆ© (šµ āˆØā„‹ š¶)) ā†’ (š¶ āˆØā„‹ š‘„) = (š¶ āˆØā„‹ (š“ āˆ© (šµ āˆØā„‹ š¶))))
8685sseq2d 3977 . . . . . . . 8 (š‘„ = (š“ āˆ© (šµ āˆØā„‹ š¶)) ā†’ (šµ āŠ† (š¶ āˆØā„‹ š‘„) ā†” šµ āŠ† (š¶ āˆØā„‹ (š“ āˆ© (šµ āˆØā„‹ š¶)))))
8784, 86anbi12d 632 . . . . . . 7 (š‘„ = (š“ āˆ© (šµ āˆØā„‹ š¶)) ā†’ ((š‘„ āŠ† š“ āˆ§ šµ āŠ† (š¶ āˆØā„‹ š‘„)) ā†” ((š“ āˆ© (šµ āˆØā„‹ š¶)) āŠ† š“ āˆ§ šµ āŠ† (š¶ āˆØā„‹ (š“ āˆ© (šµ āˆØā„‹ š¶))))))
8887rspcev 3580 . . . . . 6 (((š“ āˆ© (šµ āˆØā„‹ š¶)) āˆˆ HAtoms āˆ§ ((š“ āˆ© (šµ āˆØā„‹ š¶)) āŠ† š“ āˆ§ šµ āŠ† (š¶ āˆØā„‹ (š“ āˆ© (šµ āˆØā„‹ š¶))))) ā†’ āˆƒš‘„ āˆˆ HAtoms (š‘„ āŠ† š“ āˆ§ šµ āŠ† (š¶ āˆØā„‹ š‘„)))
8983, 88syl6 35 . . . . 5 ((šµ āˆˆ HAtoms āˆ§ š¶ āˆˆ HAtoms) ā†’ (((Ā¬ šµ = š¶ āˆ§ Ā¬ š¶ āŠ† š“) āˆ§ šµ āŠ† (š“ āˆØā„‹ š¶)) ā†’ āˆƒš‘„ āˆˆ HAtoms (š‘„ āŠ† š“ āˆ§ šµ āŠ† (š¶ āˆØā„‹ š‘„))))
9089expd 417 . . . 4 ((šµ āˆˆ HAtoms āˆ§ š¶ āˆˆ HAtoms) ā†’ ((Ā¬ šµ = š¶ āˆ§ Ā¬ š¶ āŠ† š“) ā†’ (šµ āŠ† (š“ āˆØā„‹ š¶) ā†’ āˆƒš‘„ āˆˆ HAtoms (š‘„ āŠ† š“ āˆ§ šµ āŠ† (š¶ āˆØā„‹ š‘„)))))
9143, 90biimtrid 241 . . 3 ((šµ āˆˆ HAtoms āˆ§ š¶ āˆˆ HAtoms) ā†’ (Ā¬ (šµ = š¶ āˆØ š¶ āŠ† š“) ā†’ (šµ āŠ† (š“ āˆØā„‹ š¶) ā†’ āˆƒš‘„ āˆˆ HAtoms (š‘„ āŠ† š“ āˆ§ šµ āŠ† (š¶ āˆØā„‹ š‘„)))))
9242, 91syl7 74 . 2 ((šµ āˆˆ HAtoms āˆ§ š¶ āˆˆ HAtoms) ā†’ (Ā¬ (šµ = š¶ āˆØ š¶ āŠ† š“) ā†’ ((š“ ā‰  0ā„‹ āˆ§ šµ āŠ† (š“ āˆØā„‹ š¶)) ā†’ āˆƒš‘„ āˆˆ HAtoms (š‘„ āŠ† š“ āˆ§ šµ āŠ† (š¶ āˆØā„‹ š‘„)))))
9319, 41, 92ecase3d 1033 1 ((šµ āˆˆ HAtoms āˆ§ š¶ āˆˆ HAtoms) ā†’ ((š“ ā‰  0ā„‹ āˆ§ šµ āŠ† (š“ āˆØā„‹ š¶)) ā†’ āˆƒš‘„ āˆˆ HAtoms (š‘„ āŠ† š“ āˆ§ šµ āŠ† (š¶ āˆØā„‹ š‘„))))
Colors of variables: wff setvar class
Syntax hints:  Ā¬ wn 3   ā†’ wi 4   ā†” wb 205   āˆ§ wa 397   āˆØ wo 846   āˆ§ w3a 1088   = wceq 1542   āˆˆ wcel 2107   ā‰  wne 2940  āˆƒwrex 3070   āˆ© cin 3910   āŠ† wss 3911  (class class class)co 7358   Cā„‹ cch 29913   āˆØā„‹ chj 29917  0ā„‹c0h 29919  HAtomscat 29949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-inf2 9582  ax-cc 10376  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133  ax-pre-sup 11134  ax-addf 11135  ax-mulf 11136  ax-hilex 29983  ax-hfvadd 29984  ax-hvcom 29985  ax-hvass 29986  ax-hv0cl 29987  ax-hvaddid 29988  ax-hfvmul 29989  ax-hvmulid 29990  ax-hvmulass 29991  ax-hvdistr1 29992  ax-hvdistr2 29993  ax-hvmul0 29994  ax-hfi 30063  ax-his1 30066  ax-his2 30067  ax-his3 30068  ax-his4 30069  ax-hcompl 30186
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-iin 4958  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7618  df-om 7804  df-1st 7922  df-2nd 7923  df-supp 8094  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-2o 8414  df-oadd 8417  df-omul 8418  df-er 8651  df-map 8770  df-pm 8771  df-ixp 8839  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-fsupp 9309  df-fi 9352  df-sup 9383  df-inf 9384  df-oi 9451  df-card 9880  df-acn 9883  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-div 11818  df-nn 12159  df-2 12221  df-3 12222  df-4 12223  df-5 12224  df-6 12225  df-7 12226  df-8 12227  df-9 12228  df-n0 12419  df-z 12505  df-dec 12624  df-uz 12769  df-q 12879  df-rp 12921  df-xneg 13038  df-xadd 13039  df-xmul 13040  df-ioo 13274  df-ico 13276  df-icc 13277  df-fz 13431  df-fzo 13574  df-fl 13703  df-seq 13913  df-exp 13974  df-hash 14237  df-cj 14990  df-re 14991  df-im 14992  df-sqrt 15126  df-abs 15127  df-clim 15376  df-rlim 15377  df-sum 15577  df-struct 17024  df-sets 17041  df-slot 17059  df-ndx 17071  df-base 17089  df-ress 17118  df-plusg 17151  df-mulr 17152  df-starv 17153  df-sca 17154  df-vsca 17155  df-ip 17156  df-tset 17157  df-ple 17158  df-ds 17160  df-unif 17161  df-hom 17162  df-cco 17163  df-rest 17309  df-topn 17310  df-0g 17328  df-gsum 17329  df-topgen 17330  df-pt 17331  df-prds 17334  df-xrs 17389  df-qtop 17394  df-imas 17395  df-xps 17397  df-mre 17471  df-mrc 17472  df-acs 17474  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-submnd 18607  df-mulg 18878  df-cntz 19102  df-cmn 19569  df-psmet 20804  df-xmet 20805  df-met 20806  df-bl 20807  df-mopn 20808  df-fbas 20809  df-fg 20810  df-cnfld 20813  df-top 22259  df-topon 22276  df-topsp 22298  df-bases 22312  df-cld 22386  df-ntr 22387  df-cls 22388  df-nei 22465  df-cn 22594  df-cnp 22595  df-lm 22596  df-haus 22682  df-tx 22929  df-hmeo 23122  df-fil 23213  df-fm 23305  df-flim 23306  df-flf 23307  df-xms 23689  df-ms 23690  df-tms 23691  df-cfil 24635  df-cau 24636  df-cmet 24637  df-grpo 29477  df-gid 29478  df-ginv 29479  df-gdiv 29480  df-ablo 29529  df-vc 29543  df-nv 29576  df-va 29579  df-ba 29580  df-sm 29581  df-0v 29582  df-vs 29583  df-nmcv 29584  df-ims 29585  df-dip 29685  df-ssp 29706  df-ph 29797  df-cbn 29847  df-hnorm 29952  df-hba 29953  df-hvsub 29955  df-hlim 29956  df-hcau 29957  df-sh 30191  df-ch 30205  df-oc 30236  df-ch0 30237  df-shs 30292  df-span 30293  df-chj 30294  df-chsup 30295  df-pjh 30379  df-cv 31263  df-at 31322
This theorem is referenced by:  mdsymlem3  31389
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