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Theorem brsset 31673
Description: For sets, the SSet binary relationship 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 31672 . . 3 Rel SSet
21brrelexi 5123 . 2 (𝐴 SSet 𝐵𝐴 ∈ V)
3 brsset.1 . . 3 𝐵 ∈ V
43ssex 4767 . 2 (𝐴𝐵𝐴 ∈ V)
5 breq1 4621 . . 3 (𝑥 = 𝐴 → (𝑥 SSet 𝐵𝐴 SSet 𝐵))
6 sseq1 3610 . . 3 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
7 opex 4898 . . . . . . 7 𝑥, 𝐵⟩ ∈ V
87elrn 5331 . . . . . 6 (⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )) ↔ ∃𝑦 𝑦( E ⊗ (V ∖ E ))⟨𝑥, 𝐵⟩)
9 vex 3192 . . . . . . . . 9 𝑦 ∈ V
10 vex 3192 . . . . . . . . 9 𝑥 ∈ V
119, 10, 3brtxp 31664 . . . . . . . 8 (𝑦( E ⊗ (V ∖ E ))⟨𝑥, 𝐵⟩ ↔ (𝑦 E 𝑥𝑦(V ∖ E )𝐵))
12 epel 4993 . . . . . . . . 9 (𝑦 E 𝑥𝑦𝑥)
13 brv 31661 . . . . . . . . . . 11 𝑦V𝐵
14 brdif 4670 . . . . . . . . . . 11 (𝑦(V ∖ E )𝐵 ↔ (𝑦V𝐵 ∧ ¬ 𝑦 E 𝐵))
1513, 14mpbiran 952 . . . . . . . . . 10 (𝑦(V ∖ E )𝐵 ↔ ¬ 𝑦 E 𝐵)
163epelc 4992 . . . . . . . . . 10 (𝑦 E 𝐵𝑦𝐵)
1715, 16xchbinx 324 . . . . . . . . 9 (𝑦(V ∖ E )𝐵 ↔ ¬ 𝑦𝐵)
1812, 17anbi12i 732 . . . . . . . 8 ((𝑦 E 𝑥𝑦(V ∖ E )𝐵) ↔ (𝑦𝑥 ∧ ¬ 𝑦𝐵))
1911, 18bitri 264 . . . . . . 7 (𝑦( E ⊗ (V ∖ E ))⟨𝑥, 𝐵⟩ ↔ (𝑦𝑥 ∧ ¬ 𝑦𝐵))
2019exbii 1771 . . . . . 6 (∃𝑦 𝑦( E ⊗ (V ∖ E ))⟨𝑥, 𝐵⟩ ↔ ∃𝑦(𝑦𝑥 ∧ ¬ 𝑦𝐵))
21 exanali 1783 . . . . . 6 (∃𝑦(𝑦𝑥 ∧ ¬ 𝑦𝐵) ↔ ¬ ∀𝑦(𝑦𝑥𝑦𝐵))
228, 20, 213bitrri 287 . . . . 5 (¬ ∀𝑦(𝑦𝑥𝑦𝐵) ↔ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )))
2322con1bii 346 . . . 4 (¬ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )) ↔ ∀𝑦(𝑦𝑥𝑦𝐵))
24 df-br 4619 . . . . 5 (𝑥 SSet 𝐵 ↔ ⟨𝑥, 𝐵⟩ ∈ SSet )
25 df-sset 31639 . . . . . . 7 SSet = ((V × V) ∖ ran ( E ⊗ (V ∖ E )))
2625eleq2i 2690 . . . . . 6 (⟨𝑥, 𝐵⟩ ∈ SSet ↔ ⟨𝑥, 𝐵⟩ ∈ ((V × V) ∖ ran ( E ⊗ (V ∖ E ))))
2710, 3opelvv 5131 . . . . . . 7 𝑥, 𝐵⟩ ∈ (V × V)
28 eldif 3569 . . . . . . 7 (⟨𝑥, 𝐵⟩ ∈ ((V × V) ∖ ran ( E ⊗ (V ∖ E ))) ↔ (⟨𝑥, 𝐵⟩ ∈ (V × V) ∧ ¬ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E ))))
2927, 28mpbiran 952 . . . . . 6 (⟨𝑥, 𝐵⟩ ∈ ((V × V) ∖ ran ( E ⊗ (V ∖ E ))) ↔ ¬ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )))
3026, 29bitri 264 . . . . 5 (⟨𝑥, 𝐵⟩ ∈ SSet ↔ ¬ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )))
3124, 30bitri 264 . . . 4 (𝑥 SSet 𝐵 ↔ ¬ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )))
32 dfss2 3576 . . . 4 (𝑥𝐵 ↔ ∀𝑦(𝑦𝑥𝑦𝐵))
3323, 31, 323bitr4i 292 . . 3 (𝑥 SSet 𝐵𝑥𝐵)
345, 6, 33vtoclbg 3256 . 2 (𝐴 ∈ V → (𝐴 SSet 𝐵𝐴𝐵))
352, 4, 34pm5.21nii 368 1 (𝐴 SSet 𝐵𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  wal 1478  wex 1701  wcel 1987  Vcvv 3189  cdif 3556  wss 3559  cop 4159   class class class wbr 4618   E cep 4988   × cxp 5077  ran crn 5080  ctxp 31613   SSet csset 31615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-eprel 4990  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fo 5858  df-fv 5860  df-1st 7120  df-2nd 7121  df-txp 31637  df-sset 31639
This theorem is referenced by:  idsset  31674  dfon3  31676  imagesset  31737
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