Theorem brsset 34200

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 𝐵 ↔ 𝐴 ⊆ 𝐵)

Proof of Theorem brsset

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relsset 34199	. . 3 ⊢ Rel SSet
2	1	brrelex1i 5644	. 2 ⊢ (𝐴 SSet 𝐵 → 𝐴 ∈ V)
3		brsset.1	. . 3 ⊢ 𝐵 ∈ V
4	3	ssex 5249	. 2 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)
5		breq1 5082	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 SSet 𝐵 ↔ 𝐴 SSet 𝐵))
6		sseq1 3951	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐵))
7		opex 5383	. . . . . . 7 ⊢ ⟨𝑥, 𝐵⟩ ∈ V
8	7	elrn 5801	. . . . . 6 ⊢ (⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )) ↔ ∃𝑦 𝑦( E ⊗ (V ∖ E ))⟨𝑥, 𝐵⟩)
9		vex 3435	. . . . . . . . 9 ⊢ 𝑦 ∈ V
10		vex 3435	. . . . . . . . 9 ⊢ 𝑥 ∈ V
11	9, 10, 3	brtxp 34191	. . . . . . . 8 ⊢ (𝑦( E ⊗ (V ∖ E ))⟨𝑥, 𝐵⟩ ↔ (𝑦 E 𝑥 ∧ 𝑦(V ∖ E )𝐵))
12		epel 5499	. . . . . . . . 9 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
13		brv 5391	. . . . . . . . . . 11 ⊢ 𝑦V𝐵
14		brdif 5132	. . . . . . . . . . 11 ⊢ (𝑦(V ∖ E )𝐵 ↔ (𝑦V𝐵 ∧ ¬ 𝑦 E 𝐵))
15	13, 14	mpbiran 706	. . . . . . . . . 10 ⊢ (𝑦(V ∖ E )𝐵 ↔ ¬ 𝑦 E 𝐵)
16	3	epeli 5498	. . . . . . . . . 10 ⊢ (𝑦 E 𝐵 ↔ 𝑦 ∈ 𝐵)
17	15, 16	xchbinx 334	. . . . . . . . 9 ⊢ (𝑦(V ∖ E )𝐵 ↔ ¬ 𝑦 ∈ 𝐵)
18	12, 17	anbi12i 627	. . . . . . . 8 ⊢ ((𝑦 E 𝑥 ∧ 𝑦(V ∖ E )𝐵) ↔ (𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ 𝐵))
19	11, 18	bitri 274	. . . . . . 7 ⊢ (𝑦( E ⊗ (V ∖ E ))⟨𝑥, 𝐵⟩ ↔ (𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ 𝐵))
20	19	exbii 1854	. . . . . 6 ⊢ (∃𝑦 𝑦( E ⊗ (V ∖ E ))⟨𝑥, 𝐵⟩ ↔ ∃𝑦(𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ 𝐵))
21		exanali 1866	. . . . . 6 ⊢ (∃𝑦(𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ 𝐵) ↔ ¬ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵))
22	8, 20, 21	3bitrri 298	. . . . 5 ⊢ (¬ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵) ↔ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )))
23	22	con1bii 357	. . . 4 ⊢ (¬ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )) ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵))
24		df-br 5080	. . . . 5 ⊢ (𝑥 SSet 𝐵 ↔ ⟨𝑥, 𝐵⟩ ∈ SSet )
25		df-sset 34167	. . . . . . 7 ⊢ SSet = ((V × V) ∖ ran ( E ⊗ (V ∖ E )))
26	25	eleq2i 2832	. . . . . 6 ⊢ (⟨𝑥, 𝐵⟩ ∈ SSet ↔ ⟨𝑥, 𝐵⟩ ∈ ((V × V) ∖ ran ( E ⊗ (V ∖ E ))))
27	10, 3	opelvv 5629	. . . . . . 7 ⊢ ⟨𝑥, 𝐵⟩ ∈ (V × V)
28		eldif 3902	. . . . . . 7 ⊢ (⟨𝑥, 𝐵⟩ ∈ ((V × V) ∖ ran ( E ⊗ (V ∖ E ))) ↔ (⟨𝑥, 𝐵⟩ ∈ (V × V) ∧ ¬ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E ))))
29	27, 28	mpbiran 706	. . . . . 6 ⊢ (⟨𝑥, 𝐵⟩ ∈ ((V × V) ∖ ran ( E ⊗ (V ∖ E ))) ↔ ¬ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )))
30	26, 29	bitri 274	. . . . 5 ⊢ (⟨𝑥, 𝐵⟩ ∈ SSet ↔ ¬ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )))
31	24, 30	bitri 274	. . . 4 ⊢ (𝑥 SSet 𝐵 ↔ ¬ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )))
32		dfss2 3912	. . . 4 ⊢ (𝑥 ⊆ 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵))
33	23, 31, 32	3bitr4i 303	. . 3 ⊢ (𝑥 SSet 𝐵 ↔ 𝑥 ⊆ 𝐵)
34	5, 6, 33	vtoclbg 3506	. 2 ⊢ (𝐴 ∈ V → (𝐴 SSet 𝐵 ↔ 𝐴 ⊆ 𝐵))
35	2, 4, 34	pm5.21nii 380	1 ⊢ (𝐴 SSet 𝐵 ↔ 𝐴 ⊆ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1540  ∃wex 1786   ∈ wcel 2110  Vcvv 3431   ∖ cdif 3889   ⊆ wss 3892  ⟨cop 4573   class class class wbr 5079   E cep 5495   × cxp 5588  ran crn 5591   ⊗ ctxp 34141   SSet csset 34143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356  ax-un 7583

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-eprel 5496  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-fo 6438  df-fv 6440  df-1st 7825  df-2nd 7826  df-txp 34165  df-sset 34167

This theorem is referenced by:  idsset  34201  dfon3  34203  imagesset  34264