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Theorem brsset 35870
Description: For sets, the SSet binary relation is equivalent to the subset relationship. (Contributed by Scott Fenton, 31-Mar-2012.)
Hypothesis
Ref Expression
brsset.1 𝐵 ∈ V
Assertion
Ref Expression
brsset (𝐴 SSet 𝐵𝐴𝐵)

Proof of Theorem brsset
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relsset 35869 . . 3 Rel SSet
21brrelex1i 5744 . 2 (𝐴 SSet 𝐵𝐴 ∈ V)
3 brsset.1 . . 3 𝐵 ∈ V
43ssex 5326 . 2 (𝐴𝐵𝐴 ∈ V)
5 breq1 5150 . . 3 (𝑥 = 𝐴 → (𝑥 SSet 𝐵𝐴 SSet 𝐵))
6 sseq1 4020 . . 3 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
7 opex 5474 . . . . . . 7 𝑥, 𝐵⟩ ∈ V
87elrn 5906 . . . . . 6 (⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )) ↔ ∃𝑦 𝑦( E ⊗ (V ∖ E ))⟨𝑥, 𝐵⟩)
9 vex 3481 . . . . . . . . 9 𝑦 ∈ V
10 vex 3481 . . . . . . . . 9 𝑥 ∈ V
119, 10, 3brtxp 35861 . . . . . . . 8 (𝑦( E ⊗ (V ∖ E ))⟨𝑥, 𝐵⟩ ↔ (𝑦 E 𝑥𝑦(V ∖ E )𝐵))
12 epel 5591 . . . . . . . . 9 (𝑦 E 𝑥𝑦𝑥)
13 brv 5482 . . . . . . . . . . 11 𝑦V𝐵
14 brdif 5200 . . . . . . . . . . 11 (𝑦(V ∖ E )𝐵 ↔ (𝑦V𝐵 ∧ ¬ 𝑦 E 𝐵))
1513, 14mpbiran 709 . . . . . . . . . 10 (𝑦(V ∖ E )𝐵 ↔ ¬ 𝑦 E 𝐵)
163epeli 5590 . . . . . . . . . 10 (𝑦 E 𝐵𝑦𝐵)
1715, 16xchbinx 334 . . . . . . . . 9 (𝑦(V ∖ E )𝐵 ↔ ¬ 𝑦𝐵)
1812, 17anbi12i 628 . . . . . . . 8 ((𝑦 E 𝑥𝑦(V ∖ E )𝐵) ↔ (𝑦𝑥 ∧ ¬ 𝑦𝐵))
1911, 18bitri 275 . . . . . . 7 (𝑦( E ⊗ (V ∖ E ))⟨𝑥, 𝐵⟩ ↔ (𝑦𝑥 ∧ ¬ 𝑦𝐵))
2019exbii 1844 . . . . . 6 (∃𝑦 𝑦( E ⊗ (V ∖ E ))⟨𝑥, 𝐵⟩ ↔ ∃𝑦(𝑦𝑥 ∧ ¬ 𝑦𝐵))
21 exanali 1856 . . . . . 6 (∃𝑦(𝑦𝑥 ∧ ¬ 𝑦𝐵) ↔ ¬ ∀𝑦(𝑦𝑥𝑦𝐵))
228, 20, 213bitrri 298 . . . . 5 (¬ ∀𝑦(𝑦𝑥𝑦𝐵) ↔ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )))
2322con1bii 356 . . . 4 (¬ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )) ↔ ∀𝑦(𝑦𝑥𝑦𝐵))
24 df-br 5148 . . . . 5 (𝑥 SSet 𝐵 ↔ ⟨𝑥, 𝐵⟩ ∈ SSet )
25 df-sset 35837 . . . . . . 7 SSet = ((V × V) ∖ ran ( E ⊗ (V ∖ E )))
2625eleq2i 2830 . . . . . 6 (⟨𝑥, 𝐵⟩ ∈ SSet ↔ ⟨𝑥, 𝐵⟩ ∈ ((V × V) ∖ ran ( E ⊗ (V ∖ E ))))
2710, 3opelvv 5728 . . . . . . 7 𝑥, 𝐵⟩ ∈ (V × V)
28 eldif 3972 . . . . . . 7 (⟨𝑥, 𝐵⟩ ∈ ((V × V) ∖ ran ( E ⊗ (V ∖ E ))) ↔ (⟨𝑥, 𝐵⟩ ∈ (V × V) ∧ ¬ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E ))))
2927, 28mpbiran 709 . . . . . 6 (⟨𝑥, 𝐵⟩ ∈ ((V × V) ∖ ran ( E ⊗ (V ∖ E ))) ↔ ¬ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )))
3026, 29bitri 275 . . . . 5 (⟨𝑥, 𝐵⟩ ∈ SSet ↔ ¬ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )))
3124, 30bitri 275 . . . 4 (𝑥 SSet 𝐵 ↔ ¬ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )))
32 df-ss 3979 . . . 4 (𝑥𝐵 ↔ ∀𝑦(𝑦𝑥𝑦𝐵))
3323, 31, 323bitr4i 303 . . 3 (𝑥 SSet 𝐵𝑥𝐵)
345, 6, 33vtoclbg 3556 . 2 (𝐴 ∈ V → (𝐴 SSet 𝐵𝐴𝐵))
352, 4, 34pm5.21nii 378 1 (𝐴 SSet 𝐵𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wal 1534  wex 1775  wcel 2105  Vcvv 3477  cdif 3959  wss 3962  cop 4636   class class class wbr 5147   E cep 5587   × cxp 5686  ran crn 5689  ctxp 35811   SSet csset 35813
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-eprel 5588  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fo 6568  df-fv 6570  df-1st 8012  df-2nd 8013  df-txp 35835  df-sset 35837
This theorem is referenced by:  idsset  35871  dfon3  35873  imagesset  35934
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