Theorem brsset 36081

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 36080	. . 3 ⊢ Rel SSet
2	1	brrelex1i 5680	. 2 ⊢ (𝐴 SSet 𝐵 → 𝐴 ∈ V)
3		brsset.1	. . 3 ⊢ 𝐵 ∈ V
4	3	ssex 5266	. 2 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)
5		breq1 5101	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 SSet 𝐵 ↔ 𝐴 SSet 𝐵))
6		sseq1 3959	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐵))
7		opex 5412	. . . . . . 7 ⊢ ⟨𝑥, 𝐵⟩ ∈ V
8	7	elrn 5842	. . . . . 6 ⊢ (⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )) ↔ ∃𝑦 𝑦( E ⊗ (V ∖ E ))⟨𝑥, 𝐵⟩)
9		vex 3444	. . . . . . . . 9 ⊢ 𝑦 ∈ V
10		vex 3444	. . . . . . . . 9 ⊢ 𝑥 ∈ V
11	9, 10, 3	brtxp 36072	. . . . . . . 8 ⊢ (𝑦( E ⊗ (V ∖ E ))⟨𝑥, 𝐵⟩ ↔ (𝑦 E 𝑥 ∧ 𝑦(V ∖ E )𝐵))
12		epel 5527	. . . . . . . . 9 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
13		brv 5420	. . . . . . . . . . 11 ⊢ 𝑦V𝐵
14		brdif 5151	. . . . . . . . . . 11 ⊢ (𝑦(V ∖ E )𝐵 ↔ (𝑦V𝐵 ∧ ¬ 𝑦 E 𝐵))
15	13, 14	mpbiran 709	. . . . . . . . . 10 ⊢ (𝑦(V ∖ E )𝐵 ↔ ¬ 𝑦 E 𝐵)
16	3	epeli 5526	. . . . . . . . . 10 ⊢ (𝑦 E 𝐵 ↔ 𝑦 ∈ 𝐵)
17	15, 16	xchbinx 334	. . . . . . . . 9 ⊢ (𝑦(V ∖ E )𝐵 ↔ ¬ 𝑦 ∈ 𝐵)
18	12, 17	anbi12i 628	. . . . . . . 8 ⊢ ((𝑦 E 𝑥 ∧ 𝑦(V ∖ E )𝐵) ↔ (𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ 𝐵))
19	11, 18	bitri 275	. . . . . . 7 ⊢ (𝑦( E ⊗ (V ∖ E ))⟨𝑥, 𝐵⟩ ↔ (𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ 𝐵))
20	19	exbii 1849	. . . . . 6 ⊢ (∃𝑦 𝑦( E ⊗ (V ∖ E ))⟨𝑥, 𝐵⟩ ↔ ∃𝑦(𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ 𝐵))
21		exanali 1860	. . . . . 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 356	. . . 4 ⊢ (¬ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )) ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵))
24		df-br 5099	. . . . 5 ⊢ (𝑥 SSet 𝐵 ↔ ⟨𝑥, 𝐵⟩ ∈ SSet )
25		df-sset 36048	. . . . . . 7 ⊢ SSet = ((V × V) ∖ ran ( E ⊗ (V ∖ E )))
26	25	eleq2i 2828	. . . . . 6 ⊢ (⟨𝑥, 𝐵⟩ ∈ SSet ↔ ⟨𝑥, 𝐵⟩ ∈ ((V × V) ∖ ran ( E ⊗ (V ∖ E ))))
27	10, 3	opelvv 5664	. . . . . . 7 ⊢ ⟨𝑥, 𝐵⟩ ∈ (V × V)
28		eldif 3911	. . . . . . 7 ⊢ (⟨𝑥, 𝐵⟩ ∈ ((V × V) ∖ ran ( E ⊗ (V ∖ E ))) ↔ (⟨𝑥, 𝐵⟩ ∈ (V × V) ∧ ¬ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E ))))
29	27, 28	mpbiran 709	. . . . . 6 ⊢ (⟨𝑥, 𝐵⟩ ∈ ((V × V) ∖ ran ( E ⊗ (V ∖ E ))) ↔ ¬ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )))
30	26, 29	bitri 275	. . . . 5 ⊢ (⟨𝑥, 𝐵⟩ ∈ SSet ↔ ¬ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )))
31	24, 30	bitri 275	. . . 4 ⊢ (𝑥 SSet 𝐵 ↔ ¬ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )))
32		df-ss 3918	. . . 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 3514	. 2 ⊢ (𝐴 ∈ V → (𝐴 SSet 𝐵 ↔ 𝐴 ⊆ 𝐵))
35	2, 4, 34	pm5.21nii 378	1 ⊢ (𝐴 SSet 𝐵 ↔ 𝐴 ⊆ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1539  ∃wex 1780   ∈ wcel 2113  Vcvv 3440   ∖ cdif 3898   ⊆ wss 3901  ⟨cop 4586   class class class wbr 5098   E cep 5523   × cxp 5622  ran crn 5625   ⊗ ctxp 36022   SSet csset 36024

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-eprel 5524  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fo 6498  df-fv 6500  df-1st 7933  df-2nd 7934  df-txp 36046  df-sset 36048

This theorem is referenced by:  idsset  36082  dfon3  36084  imagesset  36147