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Theorem brsset 35890
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 35889 . . 3 Rel SSet
21brrelex1i 5741 . 2 (𝐴 SSet 𝐵𝐴 ∈ V)
3 brsset.1 . . 3 𝐵 ∈ V
43ssex 5321 . 2 (𝐴𝐵𝐴 ∈ V)
5 breq1 5146 . . 3 (𝑥 = 𝐴 → (𝑥 SSet 𝐵𝐴 SSet 𝐵))
6 sseq1 4009 . . 3 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
7 opex 5469 . . . . . . 7 𝑥, 𝐵⟩ ∈ V
87elrn 5904 . . . . . 6 (⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )) ↔ ∃𝑦 𝑦( E ⊗ (V ∖ E ))⟨𝑥, 𝐵⟩)
9 vex 3484 . . . . . . . . 9 𝑦 ∈ V
10 vex 3484 . . . . . . . . 9 𝑥 ∈ V
119, 10, 3brtxp 35881 . . . . . . . 8 (𝑦( E ⊗ (V ∖ E ))⟨𝑥, 𝐵⟩ ↔ (𝑦 E 𝑥𝑦(V ∖ E )𝐵))
12 epel 5587 . . . . . . . . 9 (𝑦 E 𝑥𝑦𝑥)
13 brv 5477 . . . . . . . . . . 11 𝑦V𝐵
14 brdif 5196 . . . . . . . . . . 11 (𝑦(V ∖ E )𝐵 ↔ (𝑦V𝐵 ∧ ¬ 𝑦 E 𝐵))
1513, 14mpbiran 709 . . . . . . . . . 10 (𝑦(V ∖ E )𝐵 ↔ ¬ 𝑦 E 𝐵)
163epeli 5586 . . . . . . . . . 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 1848 . . . . . 6 (∃𝑦 𝑦( E ⊗ (V ∖ E ))⟨𝑥, 𝐵⟩ ↔ ∃𝑦(𝑦𝑥 ∧ ¬ 𝑦𝐵))
21 exanali 1859 . . . . . 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 5144 . . . . 5 (𝑥 SSet 𝐵 ↔ ⟨𝑥, 𝐵⟩ ∈ SSet )
25 df-sset 35857 . . . . . . 7 SSet = ((V × V) ∖ ran ( E ⊗ (V ∖ E )))
2625eleq2i 2833 . . . . . 6 (⟨𝑥, 𝐵⟩ ∈ SSet ↔ ⟨𝑥, 𝐵⟩ ∈ ((V × V) ∖ ran ( E ⊗ (V ∖ E ))))
2710, 3opelvv 5725 . . . . . . 7 𝑥, 𝐵⟩ ∈ (V × V)
28 eldif 3961 . . . . . . 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 3968 . . . 4 (𝑥𝐵 ↔ ∀𝑦(𝑦𝑥𝑦𝐵))
3323, 31, 323bitr4i 303 . . 3 (𝑥 SSet 𝐵𝑥𝐵)
345, 6, 33vtoclbg 3557 . 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 1538  wex 1779  wcel 2108  Vcvv 3480  cdif 3948  wss 3951  cop 4632   class class class wbr 5143   E cep 5583   × cxp 5683  ran crn 5686  ctxp 35831   SSet csset 35833
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-eprel 5584  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569  df-1st 8014  df-2nd 8015  df-txp 35855  df-sset 35857
This theorem is referenced by:  idsset  35891  dfon3  35893  imagesset  35954
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