Theorem canthwe 9352

Description: The set of well-orders of a set 𝐴 strictly dominates 𝐴. A stronger form of canth2 7998. Corollary 1.4(b) of [KanamoriPincus] p. 417. (Contributed by Mario Carneiro, 31-May-2015.)

Hypothesis

Ref	Expression
canthwe.1	⊢ 𝑂 = {⟨𝑥, 𝑟⟩ ∣ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)}

Assertion

Ref	Expression
canthwe	⊢ (𝐴 ∈ 𝑉 → 𝐴 ≺ 𝑂)

Distinct variable groups:   𝑥,𝑟,𝑂   𝑉,𝑟,𝑥   𝐴,𝑟,𝑥

Proof of Theorem canthwe

Dummy variables 𝑢 𝑦 𝑓 𝑣 𝑤 𝑎 𝑠 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1054	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑥 ⊆ 𝐴)
2		selpw 4115	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
3	1, 2	sylibr 223	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑥 ∈ 𝒫 𝐴)
4		simp2 1055	. . . . . . . . 9 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑟 ⊆ (𝑥 × 𝑥))
5		xpss12 5148	. . . . . . . . . 10 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐴) → (𝑥 × 𝑥) ⊆ (𝐴 × 𝐴))
6	1, 1, 5	syl2anc 691	. . . . . . . . 9 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → (𝑥 × 𝑥) ⊆ (𝐴 × 𝐴))
7	4, 6	sstrd 3578	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑟 ⊆ (𝐴 × 𝐴))
8		selpw 4115	. . . . . . . 8 ⊢ (𝑟 ∈ 𝒫 (𝐴 × 𝐴) ↔ 𝑟 ⊆ (𝐴 × 𝐴))
9	7, 8	sylibr 223	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑟 ∈ 𝒫 (𝐴 × 𝐴))
10	3, 9	jca 553	. . . . . 6 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → (𝑥 ∈ 𝒫 𝐴 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐴)))
11	10	ssopab2i 4928	. . . . 5 ⊢ {⟨𝑥, 𝑟⟩ ∣ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)} ⊆ {⟨𝑥, 𝑟⟩ ∣ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐴))}
12		canthwe.1	. . . . 5 ⊢ 𝑂 = {⟨𝑥, 𝑟⟩ ∣ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)}
13		df-xp 5044	. . . . 5 ⊢ (𝒫 𝐴 × 𝒫 (𝐴 × 𝐴)) = {⟨𝑥, 𝑟⟩ ∣ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐴))}
14	11, 12, 13	3sstr4i 3607	. . . 4 ⊢ 𝑂 ⊆ (𝒫 𝐴 × 𝒫 (𝐴 × 𝐴))
15		pwexg 4776	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
16		sqxpexg 6861	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × 𝐴) ∈ V)
17		pwexg 4776	. . . . . 6 ⊢ ((𝐴 × 𝐴) ∈ V → 𝒫 (𝐴 × 𝐴) ∈ V)
18	16, 17	syl 17	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝒫 (𝐴 × 𝐴) ∈ V)
19		xpexg 6858	. . . . 5 ⊢ ((𝒫 𝐴 ∈ V ∧ 𝒫 (𝐴 × 𝐴) ∈ V) → (𝒫 𝐴 × 𝒫 (𝐴 × 𝐴)) ∈ V)
20	15, 18, 19	syl2anc 691	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 × 𝒫 (𝐴 × 𝐴)) ∈ V)
21		ssexg 4732	. . . 4 ⊢ ((𝑂 ⊆ (𝒫 𝐴 × 𝒫 (𝐴 × 𝐴)) ∧ (𝒫 𝐴 × 𝒫 (𝐴 × 𝐴)) ∈ V) → 𝑂 ∈ V)
22	14, 20, 21	sylancr 694	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝑂 ∈ V)
23		simpr 476	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑢 ∈ 𝐴) → 𝑢 ∈ 𝐴)
24	23	snssd 4281	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑢 ∈ 𝐴) → {𝑢} ⊆ 𝐴)
25		0ss 3924	. . . . . . . 8 ⊢ ∅ ⊆ ({𝑢} × {𝑢})
26	25	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑢 ∈ 𝐴) → ∅ ⊆ ({𝑢} × {𝑢}))
27		rel0 5166	. . . . . . . 8 ⊢ Rel ∅
28		br0 4631	. . . . . . . . 9 ⊢ ¬ 𝑢∅𝑢
29		wesn 5113	. . . . . . . . 9 ⊢ (Rel ∅ → (∅ We {𝑢} ↔ ¬ 𝑢∅𝑢))
30	28, 29	mpbiri 247	. . . . . . . 8 ⊢ (Rel ∅ → ∅ We {𝑢})
31	27, 30	mp1i 13	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑢 ∈ 𝐴) → ∅ We {𝑢})
32		snex 4835	. . . . . . . 8 ⊢ {𝑢} ∈ V
33		0ex 4718	. . . . . . . 8 ⊢ ∅ ∈ V
34		simpl 472	. . . . . . . . . 10 ⊢ ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → 𝑥 = {𝑢})
35	34	sseq1d 3595	. . . . . . . . 9 ⊢ ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → (𝑥 ⊆ 𝐴 ↔ {𝑢} ⊆ 𝐴))
36		simpr 476	. . . . . . . . . 10 ⊢ ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → 𝑟 = ∅)
37	34	sqxpeqd 5065	. . . . . . . . . 10 ⊢ ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → (𝑥 × 𝑥) = ({𝑢} × {𝑢}))
38	36, 37	sseq12d 3597	. . . . . . . . 9 ⊢ ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → (𝑟 ⊆ (𝑥 × 𝑥) ↔ ∅ ⊆ ({𝑢} × {𝑢})))
39		weeq2 5027	. . . . . . . . . 10 ⊢ (𝑥 = {𝑢} → (𝑟 We 𝑥 ↔ 𝑟 We {𝑢}))
40		weeq1 5026	. . . . . . . . . 10 ⊢ (𝑟 = ∅ → (𝑟 We {𝑢} ↔ ∅ We {𝑢}))
41	39, 40	sylan9bb 732	. . . . . . . . 9 ⊢ ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → (𝑟 We 𝑥 ↔ ∅ We {𝑢}))
42	35, 38, 41	3anbi123d 1391	. . . . . . . 8 ⊢ ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ↔ ({𝑢} ⊆ 𝐴 ∧ ∅ ⊆ ({𝑢} × {𝑢}) ∧ ∅ We {𝑢})))
43	32, 33, 42	opelopaba 4916	. . . . . . 7 ⊢ (⟨{𝑢}, ∅⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)} ↔ ({𝑢} ⊆ 𝐴 ∧ ∅ ⊆ ({𝑢} × {𝑢}) ∧ ∅ We {𝑢}))
44	24, 26, 31, 43	syl3anbrc 1239	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑢 ∈ 𝐴) → ⟨{𝑢}, ∅⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)})
45	44, 12	syl6eleqr 2699	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑢 ∈ 𝐴) → ⟨{𝑢}, ∅⟩ ∈ 𝑂)
46	45	ex 449	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑢 ∈ 𝐴 → ⟨{𝑢}, ∅⟩ ∈ 𝑂))
47		eqid 2610	. . . . . . 7 ⊢ ∅ = ∅
48		snex 4835	. . . . . . . 8 ⊢ {𝑣} ∈ V
49	48, 33	opth2 4875	. . . . . . 7 ⊢ (⟨{𝑢}, ∅⟩ = ⟨{𝑣}, ∅⟩ ↔ ({𝑢} = {𝑣} ∧ ∅ = ∅))
50	47, 49	mpbiran2 956	. . . . . 6 ⊢ (⟨{𝑢}, ∅⟩ = ⟨{𝑣}, ∅⟩ ↔ {𝑢} = {𝑣})
51		vex 3176	. . . . . . 7 ⊢ 𝑢 ∈ V
52		sneqbg 4314	. . . . . . 7 ⊢ (𝑢 ∈ V → ({𝑢} = {𝑣} ↔ 𝑢 = 𝑣))
53	51, 52	ax-mp 5	. . . . . 6 ⊢ ({𝑢} = {𝑣} ↔ 𝑢 = 𝑣)
54	50, 53	bitri 263	. . . . 5 ⊢ (⟨{𝑢}, ∅⟩ = ⟨{𝑣}, ∅⟩ ↔ 𝑢 = 𝑣)
55	54	2a1i 12	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (⟨{𝑢}, ∅⟩ = ⟨{𝑣}, ∅⟩ ↔ 𝑢 = 𝑣)))
56	46, 55	dom2d 7882	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑂 ∈ V → 𝐴 ≼ 𝑂))
57	22, 56	mpd 15	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≼ 𝑂)
58		eqid 2610	. . . . . . 7 ⊢ {⟨𝑎, 𝑠⟩ ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧 ∈ 𝑎 [(◡𝑠 “ {𝑧}) / 𝑣](𝑣𝑓(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))} = {⟨𝑎, 𝑠⟩ ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧 ∈ 𝑎 [(◡𝑠 “ {𝑧}) / 𝑣](𝑣𝑓(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))}
59	58	fpwwe2cbv 9331	. . . . . 6 ⊢ {⟨𝑎, 𝑠⟩ ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧 ∈ 𝑎 [(◡𝑠 “ {𝑧}) / 𝑣](𝑣𝑓(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))} = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑤](𝑤𝑓(𝑟 ∩ (𝑤 × 𝑤))) = 𝑦))}
60		eqid 2610	. . . . . 6 ⊢ ∪ dom {⟨𝑎, 𝑠⟩ ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧 ∈ 𝑎 [(◡𝑠 “ {𝑧}) / 𝑣](𝑣𝑓(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))} = ∪ dom {⟨𝑎, 𝑠⟩ ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧 ∈ 𝑎 [(◡𝑠 “ {𝑧}) / 𝑣](𝑣𝑓(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))}
61		eqid 2610	. . . . . 6 ⊢ (◡({⟨𝑎, 𝑠⟩ ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧 ∈ 𝑎 [(◡𝑠 “ {𝑧}) / 𝑣](𝑣𝑓(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))}‘∪ dom {⟨𝑎, 𝑠⟩ ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧 ∈ 𝑎 [(◡𝑠 “ {𝑧}) / 𝑣](𝑣𝑓(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))}) “ {(∪ dom {⟨𝑎, 𝑠⟩ ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧 ∈ 𝑎 [(◡𝑠 “ {𝑧}) / 𝑣](𝑣𝑓(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))}𝑓({⟨𝑎, 𝑠⟩ ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧 ∈ 𝑎 [(◡𝑠 “ {𝑧}) / 𝑣](𝑣𝑓(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))}‘∪ dom {⟨𝑎, 𝑠⟩ ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧 ∈ 𝑎 [(◡𝑠 “ {𝑧}) / 𝑣](𝑣𝑓(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))}))}) = (◡({⟨𝑎, 𝑠⟩ ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧 ∈ 𝑎 [(◡𝑠 “ {𝑧}) / 𝑣](𝑣𝑓(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))}‘∪ dom {⟨𝑎, 𝑠⟩ ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧 ∈ 𝑎 [(◡𝑠 “ {𝑧}) / 𝑣](𝑣𝑓(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))}) “ {(∪ dom {⟨𝑎, 𝑠⟩ ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧 ∈ 𝑎 [(◡𝑠 “ {𝑧}) / 𝑣](𝑣𝑓(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))}𝑓({⟨𝑎, 𝑠⟩ ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧 ∈ 𝑎 [(◡𝑠 “ {𝑧}) / 𝑣](𝑣𝑓(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))}‘∪ dom {⟨𝑎, 𝑠⟩ ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧 ∈ 𝑎 [(◡𝑠 “ {𝑧}) / 𝑣](𝑣𝑓(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))}))})
62	12, 59, 60, 61	canthwelem 9351	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝑓:𝑂–1-1→𝐴)
63		f1of1 6049	. . . . 5 ⊢ (𝑓:𝑂–1-1-onto→𝐴 → 𝑓:𝑂–1-1→𝐴)
64	62, 63	nsyl 134	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝑓:𝑂–1-1-onto→𝐴)
65	64	nexdv 1851	. . 3 ⊢ (𝐴 ∈ 𝑉 → ¬ ∃𝑓 𝑓:𝑂–1-1-onto→𝐴)
66		ensym 7891	. . . 4 ⊢ (𝐴 ≈ 𝑂 → 𝑂 ≈ 𝐴)
67		bren 7850	. . . 4 ⊢ (𝑂 ≈ 𝐴 ↔ ∃𝑓 𝑓:𝑂–1-1-onto→𝐴)
68	66, 67	sylib 207	. . 3 ⊢ (𝐴 ≈ 𝑂 → ∃𝑓 𝑓:𝑂–1-1-onto→𝐴)
69	65, 68	nsyl 134	. 2 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝐴 ≈ 𝑂)
70		brsdom 7864	. 2 ⊢ (𝐴 ≺ 𝑂 ↔ (𝐴 ≼ 𝑂 ∧ ¬ 𝐴 ≈ 𝑂))
71	57, 69, 70	sylanbrc 695	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≺ 𝑂)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  [wsbc 3402   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  ⟨cop 4131  ∪ cuni 4372   class class class wbr 4583  {copab 4642   We wwe 4996   × cxp 5036  ^◡ccnv 5037  dom cdm 5038   “ cima 5041  Rel wrel 5043  –1-1→wf1 5801  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ≈ cen 7838   ≼ cdom 7839   ≺ csdm 7840

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-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-wrecs 7294  df-recs 7355  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-oi 8298

This theorem is referenced by: (None)