Theorem brsset 33463

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 33462	. . 3 ⊢ Rel SSet
2	1	brrelex1i 5572	. 2 ⊢ (𝐴 SSet 𝐵 → 𝐴 ∈ V)
3		brsset.1	. . 3 ⊢ 𝐵 ∈ V
4	3	ssex 5189	. 2 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)
5		breq1 5033	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 SSet 𝐵 ↔ 𝐴 SSet 𝐵))
6		sseq1 3940	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐵))
7		opex 5321	. . . . . . 7 ⊢ ⟨𝑥, 𝐵⟩ ∈ V
8	7	elrn 5786	. . . . . 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 33454	. . . . . . . 8 ⊢ (𝑦( E ⊗ (V ∖ E ))⟨𝑥, 𝐵⟩ ↔ (𝑦 E 𝑥 ∧ 𝑦(V ∖ E )𝐵))
12		epel 5433	. . . . . . . . 9 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
13		brv 5329	. . . . . . . . . . 11 ⊢ 𝑦V𝐵
14		brdif 5083	. . . . . . . . . . 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 5432	. . . . . . . . . 10 ⊢ (𝑦 E 𝐵 ↔ 𝑦 ∈ 𝐵)
17	15, 16	xchbinx 337	. . . . . . . . 9 ⊢ (𝑦(V ∖ E )𝐵 ↔ ¬ 𝑦 ∈ 𝐵)
18	12, 17	anbi12i 629	. . . . . . . 8 ⊢ ((𝑦 E 𝑥 ∧ 𝑦(V ∖ E )𝐵) ↔ (𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ 𝐵))
19	11, 18	bitri 278	. . . . . . 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 301	. . . . 5 ⊢ (¬ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵) ↔ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )))
23	22	con1bii 360	. . . 4 ⊢ (¬ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )) ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵))
24		df-br 5031	. . . . 5 ⊢ (𝑥 SSet 𝐵 ↔ ⟨𝑥, 𝐵⟩ ∈ SSet )
25		df-sset 33430	. . . . . . 7 ⊢ SSet = ((V × V) ∖ ran ( E ⊗ (V ∖ E )))
26	25	eleq2i 2881	. . . . . 6 ⊢ (⟨𝑥, 𝐵⟩ ∈ SSet ↔ ⟨𝑥, 𝐵⟩ ∈ ((V × V) ∖ ran ( E ⊗ (V ∖ E ))))
27	10, 3	opelvv 5558	. . . . . . 7 ⊢ ⟨𝑥, 𝐵⟩ ∈ (V × V)
28		eldif 3891	. . . . . . 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 278	. . . . 5 ⊢ (⟨𝑥, 𝐵⟩ ∈ SSet ↔ ¬ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )))
31	24, 30	bitri 278	. . . 4 ⊢ (𝑥 SSet 𝐵 ↔ ¬ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )))
32		dfss2 3901	. . . 4 ⊢ (𝑥 ⊆ 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵))
33	23, 31, 32	3bitr4i 306	. . 3 ⊢ (𝑥 SSet 𝐵 ↔ 𝑥 ⊆ 𝐵)
34	5, 6, 33	vtoclbg 3517	. 2 ⊢ (𝐴 ∈ V → (𝐴 SSet 𝐵 ↔ 𝐴 ⊆ 𝐵))
35	2, 4, 34	pm5.21nii 383	1 ⊢ (𝐴 SSet 𝐵 ↔ 𝐴 ⊆ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃wex 1781   ∈ wcel 2111  Vcvv 3441   ∖ cdif 3878   ⊆ wss 3881  ⟨cop 4531   class class class wbr 5030   E cep 5429   × cxp 5517  ran crn 5520   ⊗ ctxp 33404   SSet csset 33406

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-eprel 5430  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fo 6330  df-fv 6332  df-1st 7671  df-2nd 7672  df-txp 33428  df-sset 33430

This theorem is referenced by:  idsset  33464  dfon3  33466  imagesset  33527