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Theorem brsset 33345
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 33344 . . 3 Rel SSet
21brrelex1i 5603 . 2 (𝐴 SSet 𝐵𝐴 ∈ V)
3 brsset.1 . . 3 𝐵 ∈ V
43ssex 5218 . 2 (𝐴𝐵𝐴 ∈ V)
5 breq1 5062 . . 3 (𝑥 = 𝐴 → (𝑥 SSet 𝐵𝐴 SSet 𝐵))
6 sseq1 3992 . . 3 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
7 opex 5349 . . . . . . 7 𝑥, 𝐵⟩ ∈ V
87elrn 5817 . . . . . 6 (⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )) ↔ ∃𝑦 𝑦( E ⊗ (V ∖ E ))⟨𝑥, 𝐵⟩)
9 vex 3498 . . . . . . . . 9 𝑦 ∈ V
10 vex 3498 . . . . . . . . 9 𝑥 ∈ V
119, 10, 3brtxp 33336 . . . . . . . 8 (𝑦( E ⊗ (V ∖ E ))⟨𝑥, 𝐵⟩ ↔ (𝑦 E 𝑥𝑦(V ∖ E )𝐵))
12 epel 5464 . . . . . . . . 9 (𝑦 E 𝑥𝑦𝑥)
13 brv 5357 . . . . . . . . . . 11 𝑦V𝐵
14 brdif 5112 . . . . . . . . . . 11 (𝑦(V ∖ E )𝐵 ↔ (𝑦V𝐵 ∧ ¬ 𝑦 E 𝐵))
1513, 14mpbiran 707 . . . . . . . . . 10 (𝑦(V ∖ E )𝐵 ↔ ¬ 𝑦 E 𝐵)
163epeli 5463 . . . . . . . . . 10 (𝑦 E 𝐵𝑦𝐵)
1715, 16xchbinx 336 . . . . . . . . 9 (𝑦(V ∖ E )𝐵 ↔ ¬ 𝑦𝐵)
1812, 17anbi12i 628 . . . . . . . 8 ((𝑦 E 𝑥𝑦(V ∖ E )𝐵) ↔ (𝑦𝑥 ∧ ¬ 𝑦𝐵))
1911, 18bitri 277 . . . . . . 7 (𝑦( E ⊗ (V ∖ E ))⟨𝑥, 𝐵⟩ ↔ (𝑦𝑥 ∧ ¬ 𝑦𝐵))
2019exbii 1844 . . . . . 6 (∃𝑦 𝑦( E ⊗ (V ∖ E ))⟨𝑥, 𝐵⟩ ↔ ∃𝑦(𝑦𝑥 ∧ ¬ 𝑦𝐵))
21 exanali 1855 . . . . . 6 (∃𝑦(𝑦𝑥 ∧ ¬ 𝑦𝐵) ↔ ¬ ∀𝑦(𝑦𝑥𝑦𝐵))
228, 20, 213bitrri 300 . . . . 5 (¬ ∀𝑦(𝑦𝑥𝑦𝐵) ↔ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )))
2322con1bii 359 . . . 4 (¬ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )) ↔ ∀𝑦(𝑦𝑥𝑦𝐵))
24 df-br 5060 . . . . 5 (𝑥 SSet 𝐵 ↔ ⟨𝑥, 𝐵⟩ ∈ SSet )
25 df-sset 33312 . . . . . . 7 SSet = ((V × V) ∖ ran ( E ⊗ (V ∖ E )))
2625eleq2i 2904 . . . . . 6 (⟨𝑥, 𝐵⟩ ∈ SSet ↔ ⟨𝑥, 𝐵⟩ ∈ ((V × V) ∖ ran ( E ⊗ (V ∖ E ))))
2710, 3opelvv 5589 . . . . . . 7 𝑥, 𝐵⟩ ∈ (V × V)
28 eldif 3946 . . . . . . 7 (⟨𝑥, 𝐵⟩ ∈ ((V × V) ∖ ran ( E ⊗ (V ∖ E ))) ↔ (⟨𝑥, 𝐵⟩ ∈ (V × V) ∧ ¬ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E ))))
2927, 28mpbiran 707 . . . . . 6 (⟨𝑥, 𝐵⟩ ∈ ((V × V) ∖ ran ( E ⊗ (V ∖ E ))) ↔ ¬ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )))
3026, 29bitri 277 . . . . 5 (⟨𝑥, 𝐵⟩ ∈ SSet ↔ ¬ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )))
3124, 30bitri 277 . . . 4 (𝑥 SSet 𝐵 ↔ ¬ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )))
32 dfss2 3955 . . . 4 (𝑥𝐵 ↔ ∀𝑦(𝑦𝑥𝑦𝐵))
3323, 31, 323bitr4i 305 . . 3 (𝑥 SSet 𝐵𝑥𝐵)
345, 6, 33vtoclbg 3569 . 2 (𝐴 ∈ V → (𝐴 SSet 𝐵𝐴𝐵))
352, 4, 34pm5.21nii 382 1 (𝐴 SSet 𝐵𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  wal 1531  wex 1776  wcel 2110  Vcvv 3495  cdif 3933  wss 3936  cop 4567   class class class wbr 5059   E cep 5459   × cxp 5548  ran crn 5551  ctxp 33286   SSet csset 33288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-eprel 5460  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-fo 6356  df-fv 6358  df-1st 7683  df-2nd 7684  df-txp 33310  df-sset 33312
This theorem is referenced by:  idsset  33346  dfon3  33348  imagesset  33409
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