Theorem isotone1 37366

Description: Two different ways to say subset relation persists across applications of a function. (Contributed by RP, 31-May-2021.)

Assertion

Ref	Expression
isotone1	⊢ (∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴(𝑎 ⊆ 𝑏 → (𝐹‘𝑎) ⊆ (𝐹‘𝑏)) ↔ ∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘(𝑎 ∪ 𝑏)))

Distinct variable groups:   𝐴,𝑎,𝑏   𝐹,𝑎,𝑏

Proof of Theorem isotone1

Dummy variables 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseq1 3589	. . . 4 ⊢ (𝑎 = 𝑐 → (𝑎 ⊆ 𝑏 ↔ 𝑐 ⊆ 𝑏))
2		fveq2 6103	. . . . 5 ⊢ (𝑎 = 𝑐 → (𝐹‘𝑎) = (𝐹‘𝑐))
3	2	sseq1d 3595	. . . 4 ⊢ (𝑎 = 𝑐 → ((𝐹‘𝑎) ⊆ (𝐹‘𝑏) ↔ (𝐹‘𝑐) ⊆ (𝐹‘𝑏)))
4	1, 3	imbi12d 333	. . 3 ⊢ (𝑎 = 𝑐 → ((𝑎 ⊆ 𝑏 → (𝐹‘𝑎) ⊆ (𝐹‘𝑏)) ↔ (𝑐 ⊆ 𝑏 → (𝐹‘𝑐) ⊆ (𝐹‘𝑏))))
5		sseq2 3590	. . . 4 ⊢ (𝑏 = 𝑑 → (𝑐 ⊆ 𝑏 ↔ 𝑐 ⊆ 𝑑))
6		fveq2 6103	. . . . 5 ⊢ (𝑏 = 𝑑 → (𝐹‘𝑏) = (𝐹‘𝑑))
7	6	sseq2d 3596	. . . 4 ⊢ (𝑏 = 𝑑 → ((𝐹‘𝑐) ⊆ (𝐹‘𝑏) ↔ (𝐹‘𝑐) ⊆ (𝐹‘𝑑)))
8	5, 7	imbi12d 333	. . 3 ⊢ (𝑏 = 𝑑 → ((𝑐 ⊆ 𝑏 → (𝐹‘𝑐) ⊆ (𝐹‘𝑏)) ↔ (𝑐 ⊆ 𝑑 → (𝐹‘𝑐) ⊆ (𝐹‘𝑑))))
9	4, 8	cbvral2v 3155	. 2 ⊢ (∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴(𝑎 ⊆ 𝑏 → (𝐹‘𝑎) ⊆ (𝐹‘𝑏)) ↔ ∀𝑐 ∈ 𝒫 𝐴∀𝑑 ∈ 𝒫 𝐴(𝑐 ⊆ 𝑑 → (𝐹‘𝑐) ⊆ (𝐹‘𝑑)))
10		ssun1 3738	. . . . . 6 ⊢ 𝑎 ⊆ (𝑎 ∪ 𝑏)
11		simprl 790	. . . . . . 7 ⊢ ((∀𝑐 ∈ 𝒫 𝐴∀𝑑 ∈ 𝒫 𝐴(𝑐 ⊆ 𝑑 → (𝐹‘𝑐) ⊆ (𝐹‘𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → 𝑎 ∈ 𝒫 𝐴)
12		elpwi 4117	. . . . . . . . . . 11 ⊢ (𝑎 ∈ 𝒫 𝐴 → 𝑎 ⊆ 𝐴)
13	12	adantr 480	. . . . . . . . . 10 ⊢ ((𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴) → 𝑎 ⊆ 𝐴)
14		elpwi 4117	. . . . . . . . . . 11 ⊢ (𝑏 ∈ 𝒫 𝐴 → 𝑏 ⊆ 𝐴)
15	14	adantl 481	. . . . . . . . . 10 ⊢ ((𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴) → 𝑏 ⊆ 𝐴)
16	13, 15	unssd 3751	. . . . . . . . 9 ⊢ ((𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴) → (𝑎 ∪ 𝑏) ⊆ 𝐴)
17		vex 3176	. . . . . . . . . . 11 ⊢ 𝑎 ∈ V
18		vex 3176	. . . . . . . . . . 11 ⊢ 𝑏 ∈ V
19	17, 18	unex 6854	. . . . . . . . . 10 ⊢ (𝑎 ∪ 𝑏) ∈ V
20	19	elpw 4114	. . . . . . . . 9 ⊢ ((𝑎 ∪ 𝑏) ∈ 𝒫 𝐴 ↔ (𝑎 ∪ 𝑏) ⊆ 𝐴)
21	16, 20	sylibr 223	. . . . . . . 8 ⊢ ((𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴) → (𝑎 ∪ 𝑏) ∈ 𝒫 𝐴)
22	21	adantl 481	. . . . . . 7 ⊢ ((∀𝑐 ∈ 𝒫 𝐴∀𝑑 ∈ 𝒫 𝐴(𝑐 ⊆ 𝑑 → (𝐹‘𝑐) ⊆ (𝐹‘𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → (𝑎 ∪ 𝑏) ∈ 𝒫 𝐴)
23		simpl 472	. . . . . . 7 ⊢ ((∀𝑐 ∈ 𝒫 𝐴∀𝑑 ∈ 𝒫 𝐴(𝑐 ⊆ 𝑑 → (𝐹‘𝑐) ⊆ (𝐹‘𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → ∀𝑐 ∈ 𝒫 𝐴∀𝑑 ∈ 𝒫 𝐴(𝑐 ⊆ 𝑑 → (𝐹‘𝑐) ⊆ (𝐹‘𝑑)))
24		sseq1 3589	. . . . . . . . 9 ⊢ (𝑐 = 𝑎 → (𝑐 ⊆ 𝑑 ↔ 𝑎 ⊆ 𝑑))
25		fveq2 6103	. . . . . . . . . 10 ⊢ (𝑐 = 𝑎 → (𝐹‘𝑐) = (𝐹‘𝑎))
26	25	sseq1d 3595	. . . . . . . . 9 ⊢ (𝑐 = 𝑎 → ((𝐹‘𝑐) ⊆ (𝐹‘𝑑) ↔ (𝐹‘𝑎) ⊆ (𝐹‘𝑑)))
27	24, 26	imbi12d 333	. . . . . . . 8 ⊢ (𝑐 = 𝑎 → ((𝑐 ⊆ 𝑑 → (𝐹‘𝑐) ⊆ (𝐹‘𝑑)) ↔ (𝑎 ⊆ 𝑑 → (𝐹‘𝑎) ⊆ (𝐹‘𝑑))))
28		sseq2 3590	. . . . . . . . 9 ⊢ (𝑑 = (𝑎 ∪ 𝑏) → (𝑎 ⊆ 𝑑 ↔ 𝑎 ⊆ (𝑎 ∪ 𝑏)))
29		fveq2 6103	. . . . . . . . . 10 ⊢ (𝑑 = (𝑎 ∪ 𝑏) → (𝐹‘𝑑) = (𝐹‘(𝑎 ∪ 𝑏)))
30	29	sseq2d 3596	. . . . . . . . 9 ⊢ (𝑑 = (𝑎 ∪ 𝑏) → ((𝐹‘𝑎) ⊆ (𝐹‘𝑑) ↔ (𝐹‘𝑎) ⊆ (𝐹‘(𝑎 ∪ 𝑏))))
31	28, 30	imbi12d 333	. . . . . . . 8 ⊢ (𝑑 = (𝑎 ∪ 𝑏) → ((𝑎 ⊆ 𝑑 → (𝐹‘𝑎) ⊆ (𝐹‘𝑑)) ↔ (𝑎 ⊆ (𝑎 ∪ 𝑏) → (𝐹‘𝑎) ⊆ (𝐹‘(𝑎 ∪ 𝑏)))))
32	27, 31	rspc2va 3294	. . . . . . 7 ⊢ (((𝑎 ∈ 𝒫 𝐴 ∧ (𝑎 ∪ 𝑏) ∈ 𝒫 𝐴) ∧ ∀𝑐 ∈ 𝒫 𝐴∀𝑑 ∈ 𝒫 𝐴(𝑐 ⊆ 𝑑 → (𝐹‘𝑐) ⊆ (𝐹‘𝑑))) → (𝑎 ⊆ (𝑎 ∪ 𝑏) → (𝐹‘𝑎) ⊆ (𝐹‘(𝑎 ∪ 𝑏))))
33	11, 22, 23, 32	syl21anc 1317	. . . . . 6 ⊢ ((∀𝑐 ∈ 𝒫 𝐴∀𝑑 ∈ 𝒫 𝐴(𝑐 ⊆ 𝑑 → (𝐹‘𝑐) ⊆ (𝐹‘𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → (𝑎 ⊆ (𝑎 ∪ 𝑏) → (𝐹‘𝑎) ⊆ (𝐹‘(𝑎 ∪ 𝑏))))
34	10, 33	mpi 20	. . . . 5 ⊢ ((∀𝑐 ∈ 𝒫 𝐴∀𝑑 ∈ 𝒫 𝐴(𝑐 ⊆ 𝑑 → (𝐹‘𝑐) ⊆ (𝐹‘𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → (𝐹‘𝑎) ⊆ (𝐹‘(𝑎 ∪ 𝑏)))
35		ssun2 3739	. . . . . 6 ⊢ 𝑏 ⊆ (𝑎 ∪ 𝑏)
36		simprr 792	. . . . . . 7 ⊢ ((∀𝑐 ∈ 𝒫 𝐴∀𝑑 ∈ 𝒫 𝐴(𝑐 ⊆ 𝑑 → (𝐹‘𝑐) ⊆ (𝐹‘𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → 𝑏 ∈ 𝒫 𝐴)
37		sseq1 3589	. . . . . . . . 9 ⊢ (𝑐 = 𝑏 → (𝑐 ⊆ 𝑑 ↔ 𝑏 ⊆ 𝑑))
38		fveq2 6103	. . . . . . . . . 10 ⊢ (𝑐 = 𝑏 → (𝐹‘𝑐) = (𝐹‘𝑏))
39	38	sseq1d 3595	. . . . . . . . 9 ⊢ (𝑐 = 𝑏 → ((𝐹‘𝑐) ⊆ (𝐹‘𝑑) ↔ (𝐹‘𝑏) ⊆ (𝐹‘𝑑)))
40	37, 39	imbi12d 333	. . . . . . . 8 ⊢ (𝑐 = 𝑏 → ((𝑐 ⊆ 𝑑 → (𝐹‘𝑐) ⊆ (𝐹‘𝑑)) ↔ (𝑏 ⊆ 𝑑 → (𝐹‘𝑏) ⊆ (𝐹‘𝑑))))
41		sseq2 3590	. . . . . . . . 9 ⊢ (𝑑 = (𝑎 ∪ 𝑏) → (𝑏 ⊆ 𝑑 ↔ 𝑏 ⊆ (𝑎 ∪ 𝑏)))
42	29	sseq2d 3596	. . . . . . . . 9 ⊢ (𝑑 = (𝑎 ∪ 𝑏) → ((𝐹‘𝑏) ⊆ (𝐹‘𝑑) ↔ (𝐹‘𝑏) ⊆ (𝐹‘(𝑎 ∪ 𝑏))))
43	41, 42	imbi12d 333	. . . . . . . 8 ⊢ (𝑑 = (𝑎 ∪ 𝑏) → ((𝑏 ⊆ 𝑑 → (𝐹‘𝑏) ⊆ (𝐹‘𝑑)) ↔ (𝑏 ⊆ (𝑎 ∪ 𝑏) → (𝐹‘𝑏) ⊆ (𝐹‘(𝑎 ∪ 𝑏)))))
44	40, 43	rspc2va 3294	. . . . . . 7 ⊢ (((𝑏 ∈ 𝒫 𝐴 ∧ (𝑎 ∪ 𝑏) ∈ 𝒫 𝐴) ∧ ∀𝑐 ∈ 𝒫 𝐴∀𝑑 ∈ 𝒫 𝐴(𝑐 ⊆ 𝑑 → (𝐹‘𝑐) ⊆ (𝐹‘𝑑))) → (𝑏 ⊆ (𝑎 ∪ 𝑏) → (𝐹‘𝑏) ⊆ (𝐹‘(𝑎 ∪ 𝑏))))
45	36, 22, 23, 44	syl21anc 1317	. . . . . 6 ⊢ ((∀𝑐 ∈ 𝒫 𝐴∀𝑑 ∈ 𝒫 𝐴(𝑐 ⊆ 𝑑 → (𝐹‘𝑐) ⊆ (𝐹‘𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → (𝑏 ⊆ (𝑎 ∪ 𝑏) → (𝐹‘𝑏) ⊆ (𝐹‘(𝑎 ∪ 𝑏))))
46	35, 45	mpi 20	. . . . 5 ⊢ ((∀𝑐 ∈ 𝒫 𝐴∀𝑑 ∈ 𝒫 𝐴(𝑐 ⊆ 𝑑 → (𝐹‘𝑐) ⊆ (𝐹‘𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → (𝐹‘𝑏) ⊆ (𝐹‘(𝑎 ∪ 𝑏)))
47	34, 46	unssd 3751	. . . 4 ⊢ ((∀𝑐 ∈ 𝒫 𝐴∀𝑑 ∈ 𝒫 𝐴(𝑐 ⊆ 𝑑 → (𝐹‘𝑐) ⊆ (𝐹‘𝑑)) ∧ (𝑎 ∈ 𝒫 𝐴 ∧ 𝑏 ∈ 𝒫 𝐴)) → ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘(𝑎 ∪ 𝑏)))
48	47	ralrimivva 2954	. . 3 ⊢ (∀𝑐 ∈ 𝒫 𝐴∀𝑑 ∈ 𝒫 𝐴(𝑐 ⊆ 𝑑 → (𝐹‘𝑐) ⊆ (𝐹‘𝑑)) → ∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘(𝑎 ∪ 𝑏)))
49		ssequn1 3745	. . . . 5 ⊢ (𝑐 ⊆ 𝑑 ↔ (𝑐 ∪ 𝑑) = 𝑑)
50	2	uneq1d 3728	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑐 → ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) = ((𝐹‘𝑐) ∪ (𝐹‘𝑏)))
51		uneq1 3722	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑐 → (𝑎 ∪ 𝑏) = (𝑐 ∪ 𝑏))
52	51	fveq2d 6107	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑐 → (𝐹‘(𝑎 ∪ 𝑏)) = (𝐹‘(𝑐 ∪ 𝑏)))
53	50, 52	sseq12d 3597	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑐 → (((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘(𝑎 ∪ 𝑏)) ↔ ((𝐹‘𝑐) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘(𝑐 ∪ 𝑏))))
54	6	uneq2d 3729	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑑 → ((𝐹‘𝑐) ∪ (𝐹‘𝑏)) = ((𝐹‘𝑐) ∪ (𝐹‘𝑑)))
55		uneq2 3723	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑑 → (𝑐 ∪ 𝑏) = (𝑐 ∪ 𝑑))
56	55	fveq2d 6107	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑑 → (𝐹‘(𝑐 ∪ 𝑏)) = (𝐹‘(𝑐 ∪ 𝑑)))
57	54, 56	sseq12d 3597	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑑 → (((𝐹‘𝑐) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘(𝑐 ∪ 𝑏)) ↔ ((𝐹‘𝑐) ∪ (𝐹‘𝑑)) ⊆ (𝐹‘(𝑐 ∪ 𝑑))))
58	53, 57	rspc2va 3294	. . . . . . . . . 10 ⊢ (((𝑐 ∈ 𝒫 𝐴 ∧ 𝑑 ∈ 𝒫 𝐴) ∧ ∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘(𝑎 ∪ 𝑏))) → ((𝐹‘𝑐) ∪ (𝐹‘𝑑)) ⊆ (𝐹‘(𝑐 ∪ 𝑑)))
59	58	ancoms 468	. . . . . . . . 9 ⊢ ((∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘(𝑎 ∪ 𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴 ∧ 𝑑 ∈ 𝒫 𝐴)) → ((𝐹‘𝑐) ∪ (𝐹‘𝑑)) ⊆ (𝐹‘(𝑐 ∪ 𝑑)))
60	59	unssad 3752	. . . . . . . 8 ⊢ ((∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘(𝑎 ∪ 𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴 ∧ 𝑑 ∈ 𝒫 𝐴)) → (𝐹‘𝑐) ⊆ (𝐹‘(𝑐 ∪ 𝑑)))
61	60	adantr 480	. . . . . . 7 ⊢ (((∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘(𝑎 ∪ 𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴 ∧ 𝑑 ∈ 𝒫 𝐴)) ∧ (𝑐 ∪ 𝑑) = 𝑑) → (𝐹‘𝑐) ⊆ (𝐹‘(𝑐 ∪ 𝑑)))
62		fveq2 6103	. . . . . . . 8 ⊢ ((𝑐 ∪ 𝑑) = 𝑑 → (𝐹‘(𝑐 ∪ 𝑑)) = (𝐹‘𝑑))
63	62	adantl 481	. . . . . . 7 ⊢ (((∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘(𝑎 ∪ 𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴 ∧ 𝑑 ∈ 𝒫 𝐴)) ∧ (𝑐 ∪ 𝑑) = 𝑑) → (𝐹‘(𝑐 ∪ 𝑑)) = (𝐹‘𝑑))
64	61, 63	sseqtrd 3604	. . . . . 6 ⊢ (((∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘(𝑎 ∪ 𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴 ∧ 𝑑 ∈ 𝒫 𝐴)) ∧ (𝑐 ∪ 𝑑) = 𝑑) → (𝐹‘𝑐) ⊆ (𝐹‘𝑑))
65	64	ex 449	. . . . 5 ⊢ ((∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘(𝑎 ∪ 𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴 ∧ 𝑑 ∈ 𝒫 𝐴)) → ((𝑐 ∪ 𝑑) = 𝑑 → (𝐹‘𝑐) ⊆ (𝐹‘𝑑)))
66	49, 65	syl5bi 231	. . . 4 ⊢ ((∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘(𝑎 ∪ 𝑏)) ∧ (𝑐 ∈ 𝒫 𝐴 ∧ 𝑑 ∈ 𝒫 𝐴)) → (𝑐 ⊆ 𝑑 → (𝐹‘𝑐) ⊆ (𝐹‘𝑑)))
67	66	ralrimivva 2954	. . 3 ⊢ (∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘(𝑎 ∪ 𝑏)) → ∀𝑐 ∈ 𝒫 𝐴∀𝑑 ∈ 𝒫 𝐴(𝑐 ⊆ 𝑑 → (𝐹‘𝑐) ⊆ (𝐹‘𝑑)))
68	48, 67	impbii 198	. 2 ⊢ (∀𝑐 ∈ 𝒫 𝐴∀𝑑 ∈ 𝒫 𝐴(𝑐 ⊆ 𝑑 → (𝐹‘𝑐) ⊆ (𝐹‘𝑑)) ↔ ∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘(𝑎 ∪ 𝑏)))
69	9, 68	bitri 263	1 ⊢ (∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴(𝑎 ⊆ 𝑏 → (𝐹‘𝑎) ⊆ (𝐹‘𝑏)) ↔ ∀𝑎 ∈ 𝒫 𝐴∀𝑏 ∈ 𝒫 𝐴((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘(𝑎 ∪ 𝑏)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∪ cun 3538   ⊆ wss 3540  𝒫 cpw 4108  ‘cfv 5804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812

This theorem is referenced by: (None)