Mathbox for Scott Fenton < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  brsset Structured version   Visualization version   GIF version

 Description: For sets, the SSet binary relation is equivalent to the subset relationship. (Contributed by Scott Fenton, 31-Mar-2012.)
Hypothesis
Ref Expression
Assertion
Ref Expression

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relsset 32535 . . 3 Rel SSet
21brrelex1i 5394 . 2 (𝐴 SSet 𝐵𝐴 ∈ V)
3 brsset.1 . . 3 𝐵 ∈ V
43ssex 5028 . 2 (𝐴𝐵𝐴 ∈ V)
5 breq1 4877 . . 3 (𝑥 = 𝐴 → (𝑥 SSet 𝐵𝐴 SSet 𝐵))
6 sseq1 3852 . . 3 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
7 opex 5154 . . . . . . 7 𝑥, 𝐵⟩ ∈ V
87elrn 5600 . . . . . 6 (⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )) ↔ ∃𝑦 𝑦( E ⊗ (V ∖ E ))⟨𝑥, 𝐵⟩)
9 vex 3418 . . . . . . . . 9 𝑦 ∈ V
10 vex 3418 . . . . . . . . 9 𝑥 ∈ V
119, 10, 3brtxp 32527 . . . . . . . 8 (𝑦( E ⊗ (V ∖ E ))⟨𝑥, 𝐵⟩ ↔ (𝑦 E 𝑥𝑦(V ∖ E )𝐵))
12 epel 5259 . . . . . . . . 9 (𝑦 E 𝑥𝑦𝑥)
13 brv 5162 . . . . . . . . . . 11 𝑦V𝐵
14 brdif 4927 . . . . . . . . . . 11 (𝑦(V ∖ E )𝐵 ↔ (𝑦V𝐵 ∧ ¬ 𝑦 E 𝐵))
1513, 14mpbiran 702 . . . . . . . . . 10 (𝑦(V ∖ E )𝐵 ↔ ¬ 𝑦 E 𝐵)
163epeli 5258 . . . . . . . . . 10 (𝑦 E 𝐵𝑦𝐵)
1715, 16xchbinx 326 . . . . . . . . 9 (𝑦(V ∖ E )𝐵 ↔ ¬ 𝑦𝐵)
1812, 17anbi12i 622 . . . . . . . 8 ((𝑦 E 𝑥𝑦(V ∖ E )𝐵) ↔ (𝑦𝑥 ∧ ¬ 𝑦𝐵))
1911, 18bitri 267 . . . . . . 7 (𝑦( E ⊗ (V ∖ E ))⟨𝑥, 𝐵⟩ ↔ (𝑦𝑥 ∧ ¬ 𝑦𝐵))
2019exbii 1949 . . . . . 6 (∃𝑦 𝑦( E ⊗ (V ∖ E ))⟨𝑥, 𝐵⟩ ↔ ∃𝑦(𝑦𝑥 ∧ ¬ 𝑦𝐵))
21 exanali 1961 . . . . . 6 (∃𝑦(𝑦𝑥 ∧ ¬ 𝑦𝐵) ↔ ¬ ∀𝑦(𝑦𝑥𝑦𝐵))
228, 20, 213bitrri 290 . . . . 5 (¬ ∀𝑦(𝑦𝑥𝑦𝐵) ↔ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )))
2322con1bii 348 . . . 4 (¬ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )) ↔ ∀𝑦(𝑦𝑥𝑦𝐵))
24 df-br 4875 . . . . 5 (𝑥 SSet 𝐵 ↔ ⟨𝑥, 𝐵⟩ ∈ SSet )
25 df-sset 32503 . . . . . . 7 SSet = ((V × V) ∖ ran ( E ⊗ (V ∖ E )))
2625eleq2i 2899 . . . . . 6 (⟨𝑥, 𝐵⟩ ∈ SSet ↔ ⟨𝑥, 𝐵⟩ ∈ ((V × V) ∖ ran ( E ⊗ (V ∖ E ))))
2710, 3opelvv 5382 . . . . . . 7 𝑥, 𝐵⟩ ∈ (V × V)
28 eldif 3809 . . . . . . 7 (⟨𝑥, 𝐵⟩ ∈ ((V × V) ∖ ran ( E ⊗ (V ∖ E ))) ↔ (⟨𝑥, 𝐵⟩ ∈ (V × V) ∧ ¬ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E ))))
2927, 28mpbiran 702 . . . . . 6 (⟨𝑥, 𝐵⟩ ∈ ((V × V) ∖ ran ( E ⊗ (V ∖ E ))) ↔ ¬ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )))
3026, 29bitri 267 . . . . 5 (⟨𝑥, 𝐵⟩ ∈ SSet ↔ ¬ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )))
3124, 30bitri 267 . . . 4 (𝑥 SSet 𝐵 ↔ ¬ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )))
32 dfss2 3816 . . . 4 (𝑥𝐵 ↔ ∀𝑦(𝑦𝑥𝑦𝐵))
3323, 31, 323bitr4i 295 . . 3 (𝑥 SSet 𝐵𝑥𝐵)
345, 6, 33vtoclbg 3484 . 2 (𝐴 ∈ V → (𝐴 SSet 𝐵𝐴𝐵))
352, 4, 34pm5.21nii 370 1 (𝐴 SSet 𝐵𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386  ∀wal 1656  ∃wex 1880   ∈ wcel 2166  Vcvv 3415   ∖ cdif 3796   ⊆ wss 3799  ⟨cop 4404   class class class wbr 4874   E cep 5255   × cxp 5341  ran crn 5344   ⊗ ctxp 32477   SSet csset 32479 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-sbc 3664  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-eprel 5256  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-fo 6130  df-fv 6132  df-1st 7429  df-2nd 7430  df-txp 32501  df-sset 32503 This theorem is referenced by:  idsset  32537  dfon3  32539  imagesset  32600
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