Theorem brsset 35475

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 35474	. . 3 ⊢ Rel SSet
2	1	brrelex1i 5728	. 2 ⊢ (𝐴 SSet 𝐵 → 𝐴 ∈ V)
3		brsset.1	. . 3 ⊢ 𝐵 ∈ V
4	3	ssex 5315	. 2 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)
5		breq1 5145	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 SSet 𝐵 ↔ 𝐴 SSet 𝐵))
6		sseq1 4003	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐵))
7		opex 5460	. . . . . . 7 ⊢ ⟨𝑥, 𝐵⟩ ∈ V
8	7	elrn 5890	. . . . . 6 ⊢ (⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )) ↔ ∃𝑦 𝑦( E ⊗ (V ∖ E ))⟨𝑥, 𝐵⟩)
9		vex 3473	. . . . . . . . 9 ⊢ 𝑦 ∈ V
10		vex 3473	. . . . . . . . 9 ⊢ 𝑥 ∈ V
11	9, 10, 3	brtxp 35466	. . . . . . . 8 ⊢ (𝑦( E ⊗ (V ∖ E ))⟨𝑥, 𝐵⟩ ↔ (𝑦 E 𝑥 ∧ 𝑦(V ∖ E )𝐵))
12		epel 5579	. . . . . . . . 9 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
13		brv 5468	. . . . . . . . . . 11 ⊢ 𝑦V𝐵
14		brdif 5195	. . . . . . . . . . 11 ⊢ (𝑦(V ∖ E )𝐵 ↔ (𝑦V𝐵 ∧ ¬ 𝑦 E 𝐵))
15	13, 14	mpbiran 708	. . . . . . . . . 10 ⊢ (𝑦(V ∖ E )𝐵 ↔ ¬ 𝑦 E 𝐵)
16	3	epeli 5578	. . . . . . . . . 10 ⊢ (𝑦 E 𝐵 ↔ 𝑦 ∈ 𝐵)
17	15, 16	xchbinx 334	. . . . . . . . 9 ⊢ (𝑦(V ∖ E )𝐵 ↔ ¬ 𝑦 ∈ 𝐵)
18	12, 17	anbi12i 626	. . . . . . . 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 1843	. . . . . 6 ⊢ (∃𝑦 𝑦( E ⊗ (V ∖ E ))⟨𝑥, 𝐵⟩ ↔ ∃𝑦(𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ 𝐵))
21		exanali 1855	. . . . . 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 5143	. . . . 5 ⊢ (𝑥 SSet 𝐵 ↔ ⟨𝑥, 𝐵⟩ ∈ SSet )
25		df-sset 35442	. . . . . . 7 ⊢ SSet = ((V × V) ∖ ran ( E ⊗ (V ∖ E )))
26	25	eleq2i 2820	. . . . . 6 ⊢ (⟨𝑥, 𝐵⟩ ∈ SSet ↔ ⟨𝑥, 𝐵⟩ ∈ ((V × V) ∖ ran ( E ⊗ (V ∖ E ))))
27	10, 3	opelvv 5712	. . . . . . 7 ⊢ ⟨𝑥, 𝐵⟩ ∈ (V × V)
28		eldif 3954	. . . . . . 7 ⊢ (⟨𝑥, 𝐵⟩ ∈ ((V × V) ∖ ran ( E ⊗ (V ∖ E ))) ↔ (⟨𝑥, 𝐵⟩ ∈ (V × V) ∧ ¬ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E ))))
29	27, 28	mpbiran 708	. . . . . 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		dfss2 3964	. . . 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 3540	. 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 205   ∧ wa 395  ∀wal 1532  ∃wex 1774   ∈ wcel 2099  Vcvv 3469   ∖ cdif 3941   ⊆ wss 3944  ⟨cop 4630   class class class wbr 5142   E cep 5575   × cxp 5670  ran crn 5673   ⊗ ctxp 35416   SSet csset 35418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-eprel 5576  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fo 6548  df-fv 6550  df-1st 7987  df-2nd 7988  df-txp 35440  df-sset 35442

This theorem is referenced by:  idsset  35476  dfon3  35478  imagesset  35539