Theorem brsset 36198

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 36197	. . 3 ⊢ Rel SSet
2	1	brrelex1i 5699	. 2 ⊢ (𝐴 SSet 𝐵 → 𝐴 ∈ V)
3		brsset.1	. . 3 ⊢ 𝐵 ∈ V
4	3	ssex 5274	. 2 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V)
5		breq1 5100	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 SSet 𝐵 ↔ 𝐴 SSet 𝐵))
6		sseq1 3959	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐵))
7		opex 5428	. . . . . . 7 ⊢ ⟨𝑥, 𝐵⟩ ∈ V
8	7	elrn 5865	. . . . . 6 ⊢ (⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )) ↔ ∃𝑦 𝑦( E ⊗ (V ∖ E ))⟨𝑥, 𝐵⟩)
9		vex 3457	. . . . . . . . 9 ⊢ 𝑦 ∈ V
10		vex 3457	. . . . . . . . 9 ⊢ 𝑥 ∈ V
11	9, 10, 3	brtxp 36189	. . . . . . . 8 ⊢ (𝑦( E ⊗ (V ∖ E ))⟨𝑥, 𝐵⟩ ↔ (𝑦 E 𝑥 ∧ 𝑦(V ∖ E )𝐵))
12		epel 5546	. . . . . . . . 9 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
13		brv 5437	. . . . . . . . . . 11 ⊢ 𝑦V𝐵
14		brdif 5150	. . . . . . . . . . 11 ⊢ (𝑦(V ∖ E )𝐵 ↔ (𝑦V𝐵 ∧ ¬ 𝑦 E 𝐵))
15	13, 14	mpbiran 719	. . . . . . . . . 10 ⊢ (𝑦(V ∖ E )𝐵 ↔ ¬ 𝑦 E 𝐵)
16	3	epeli 5545	. . . . . . . . . 10 ⊢ (𝑦 E 𝐵 ↔ 𝑦 ∈ 𝐵)
17	15, 16	xchbinx 336	. . . . . . . . 9 ⊢ (𝑦(V ∖ E )𝐵 ↔ ¬ 𝑦 ∈ 𝐵)
18	12, 17	anbi12i 637	. . . . . . . 8 ⊢ ((𝑦 E 𝑥 ∧ 𝑦(V ∖ E )𝐵) ↔ (𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ 𝐵))
19	11, 18	bitri 277	. . . . . . 7 ⊢ (𝑦( E ⊗ (V ∖ E ))⟨𝑥, 𝐵⟩ ↔ (𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ 𝐵))
20	19	exbii 1867	. . . . . 6 ⊢ (∃𝑦 𝑦( E ⊗ (V ∖ E ))⟨𝑥, 𝐵⟩ ↔ ∃𝑦(𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ 𝐵))
21		exanali 1878	. . . . . 6 ⊢ (∃𝑦(𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ 𝐵) ↔ ¬ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵))
22	8, 20, 21	3bitrri 300	. . . . 5 ⊢ (¬ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵) ↔ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )))
23	22	con1bii 358	. . . 4 ⊢ (¬ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )) ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵))
24		df-br 5098	. . . . 5 ⊢ (𝑥 SSet 𝐵 ↔ ⟨𝑥, 𝐵⟩ ∈ SSet )
25		df-sset 36165	. . . . . . 7 ⊢ SSet = ((V × V) ∖ ran ( E ⊗ (V ∖ E )))
26	25	eleq2i 2853	. . . . . 6 ⊢ (⟨𝑥, 𝐵⟩ ∈ SSet ↔ ⟨𝑥, 𝐵⟩ ∈ ((V × V) ∖ ran ( E ⊗ (V ∖ E ))))
27	10, 3	opelvv 5683	. . . . . . 7 ⊢ ⟨𝑥, 𝐵⟩ ∈ (V × V)
28		eldif 3912	. . . . . . 7 ⊢ (⟨𝑥, 𝐵⟩ ∈ ((V × V) ∖ ran ( E ⊗ (V ∖ E ))) ↔ (⟨𝑥, 𝐵⟩ ∈ (V × V) ∧ ¬ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E ))))
29	27, 28	mpbiran 719	. . . . . 6 ⊢ (⟨𝑥, 𝐵⟩ ∈ ((V × V) ∖ ran ( E ⊗ (V ∖ E ))) ↔ ¬ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )))
30	26, 29	bitri 277	. . . . 5 ⊢ (⟨𝑥, 𝐵⟩ ∈ SSet ↔ ¬ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )))
31	24, 30	bitri 277	. . . 4 ⊢ (𝑥 SSet 𝐵 ↔ ¬ ⟨𝑥, 𝐵⟩ ∈ ran ( E ⊗ (V ∖ E )))
32		df-ss 3919	. . . 4 ⊢ (𝑥 ⊆ 𝐵 ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵))
33	23, 31, 32	3bitr4i 305	. . 3 ⊢ (𝑥 SSet 𝐵 ↔ 𝑥 ⊆ 𝐵)
34	5, 6, 33	vtoclbg 3523	. 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 208   ∧ wa 399  ∀wal 1557  ∃wex 1798   ∈ wcel 2141  Vcvv 3453   ∖ cdif 3899   ⊆ wss 3902  ⟨cop 4585   class class class wbr 5097   E cep 5542   × cxp 5641  ran crn 5644   ⊗ ctxp 36139   SSet csset 36141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-eprel 5543  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-fo 6522  df-fv 6524  df-1st 7965  df-2nd 7966  df-txp 36163  df-sset 36165

This theorem is referenced by:  idsset  36199  dfon3  36201  imagesset  36264