Theorem canthwe 10637

Description: The set of well-orders of a set 𝐴 strictly dominates 𝐴. A stronger form of canth2 9119. 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 1154	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑥 ⊆ 𝐴)
2		velpw 4568	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
3	1, 2	sylibr 237	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑥 ∈ 𝒫 𝐴)
4		simp2 1155	. . . . . . . . 9 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑟 ⊆ (𝑥 × 𝑥))
5		xpss12 5678	. . . . . . . . . 10 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐴) → (𝑥 × 𝑥) ⊆ (𝐴 × 𝐴))
6	1, 1, 5	syl2anc 595	. . . . . . . . 9 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → (𝑥 × 𝑥) ⊆ (𝐴 × 𝐴))
7	4, 6	sstrd 3948	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑟 ⊆ (𝐴 × 𝐴))
8		velpw 4568	. . . . . . . 8 ⊢ (𝑟 ∈ 𝒫 (𝐴 × 𝐴) ↔ 𝑟 ⊆ (𝐴 × 𝐴))
9	7, 8	sylibr 237	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑟 ∈ 𝒫 (𝐴 × 𝐴))
10	3, 9	jca 520	. . . . . 6 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → (𝑥 ∈ 𝒫 𝐴 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐴)))
11	10	ssopab2i 5537	. . . . 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 5669	. . . . 5 ⊢ (𝒫 𝐴 × 𝒫 (𝐴 × 𝐴)) = {⟨𝑥, 𝑟⟩ ∣ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐴))}
14	11, 12, 13	3sstr4i 3989	. . . 4 ⊢ 𝑂 ⊆ (𝒫 𝐴 × 𝒫 (𝐴 × 𝐴))
15		pwexg 5351	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
16		sqxpexg 7755	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × 𝐴) ∈ V)
17	16	pwexd 5352	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝒫 (𝐴 × 𝐴) ∈ V)
18	15, 17	xpexd 7751	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 × 𝒫 (𝐴 × 𝐴)) ∈ V)
19		ssexg 5291	. . . 4 ⊢ ((𝑂 ⊆ (𝒫 𝐴 × 𝒫 (𝐴 × 𝐴)) ∧ (𝒫 𝐴 × 𝒫 (𝐴 × 𝐴)) ∈ V) → 𝑂 ∈ V)
20	14, 18, 19	sylancr 598	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝑂 ∈ V)
21		simpr 489	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑢 ∈ 𝐴) → 𝑢 ∈ 𝐴)
22	21	snssd 4753	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑢 ∈ 𝐴) → {𝑢} ⊆ 𝐴)
23		0ss 4358	. . . . . . . 8 ⊢ ∅ ⊆ ({𝑢} × {𝑢})
24	23	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑢 ∈ 𝐴) → ∅ ⊆ ({𝑢} × {𝑢}))
25		rel0 5787	. . . . . . . 8 ⊢ Rel ∅
26		br0 5161	. . . . . . . . 9 ⊢ ¬ 𝑢∅𝑢
27		wesn 5752	. . . . . . . . 9 ⊢ (Rel ∅ → (∅ We {𝑢} ↔ ¬ 𝑢∅𝑢))
28	26, 27	mpbiri 261	. . . . . . . 8 ⊢ (Rel ∅ → ∅ We {𝑢})
29	25, 28	mp1i 14	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑢 ∈ 𝐴) → ∅ We {𝑢})
30		vsnex 5408	. . . . . . . 8 ⊢ {𝑢} ∈ V
31		0ex 5271	. . . . . . . 8 ⊢ ∅ ∈ V
32		simpl 487	. . . . . . . . . 10 ⊢ ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → 𝑥 = {𝑢})
33	32	sseq1d 3969	. . . . . . . . 9 ⊢ ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → (𝑥 ⊆ 𝐴 ↔ {𝑢} ⊆ 𝐴))
34		simpr 489	. . . . . . . . . 10 ⊢ ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → 𝑟 = ∅)
35	32	sqxpeqd 5695	. . . . . . . . . 10 ⊢ ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → (𝑥 × 𝑥) = ({𝑢} × {𝑢}))
36	34, 35	sseq12d 3971	. . . . . . . . 9 ⊢ ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → (𝑟 ⊆ (𝑥 × 𝑥) ↔ ∅ ⊆ ({𝑢} × {𝑢})))
37	34, 32	weeq12d 5652	. . . . . . . . 9 ⊢ ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → (𝑟 We 𝑥 ↔ ∅ We {𝑢}))
38	33, 36, 37	3anbi123d 1464	. . . . . . . 8 ⊢ ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ↔ ({𝑢} ⊆ 𝐴 ∧ ∅ ⊆ ({𝑢} × {𝑢}) ∧ ∅ We {𝑢})))
39	30, 31, 38	opelopaba 5522	. . . . . . 7 ⊢ (⟨{𝑢}, ∅⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)} ↔ ({𝑢} ⊆ 𝐴 ∧ ∅ ⊆ ({𝑢} × {𝑢}) ∧ ∅ We {𝑢}))
40	22, 24, 29, 39	syl3anbrc 1362	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑢 ∈ 𝐴) → ⟨{𝑢}, ∅⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)})
41	40, 12	eleqtrrdi 2874	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑢 ∈ 𝐴) → ⟨{𝑢}, ∅⟩ ∈ 𝑂)
42	41	ex 417	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑢 ∈ 𝐴 → ⟨{𝑢}, ∅⟩ ∈ 𝑂))
43		eqid 2763	. . . . . . 7 ⊢ ∅ = ∅
44		vsnex 5408	. . . . . . . 8 ⊢ {𝑣} ∈ V
45	44, 31	opth2 5464	. . . . . . 7 ⊢ (⟨{𝑢}, ∅⟩ = ⟨{𝑣}, ∅⟩ ↔ ({𝑢} = {𝑣} ∧ ∅ = ∅))
46	43, 45	mpbiran2 722	. . . . . 6 ⊢ (⟨{𝑢}, ∅⟩ = ⟨{𝑣}, ∅⟩ ↔ {𝑢} = {𝑣})
47		sneqbg 4809	. . . . . . 7 ⊢ (𝑢 ∈ V → ({𝑢} = {𝑣} ↔ 𝑢 = 𝑣))
48	47	elv 3460	. . . . . 6 ⊢ ({𝑢} = {𝑣} ↔ 𝑢 = 𝑣)
49	46, 48	bitri 278	. . . . 5 ⊢ (⟨{𝑢}, ∅⟩ = ⟨{𝑣}, ∅⟩ ↔ 𝑢 = 𝑣)
50	49	2a1i 12	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (⟨{𝑢}, ∅⟩ = ⟨{𝑣}, ∅⟩ ↔ 𝑢 = 𝑣)))
51	42, 50	dom2d 8991	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑂 ∈ V → 𝐴 ≼ 𝑂))
52	20, 51	mpd 16	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≼ 𝑂)
53		eqid 2763	. . . . . . 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))}
54	53	fpwwe2cbv 10616	. . . . . 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))}
55		eqid 2763	. . . . . 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))}
56		eqid 2763	. . . . . 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))}))})
57	12, 54, 55, 56	canthwelem 10636	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝑓:𝑂–1-1→𝐴)
58		f1of1 6821	. . . . 5 ⊢ (𝑓:𝑂–1-1-onto→𝐴 → 𝑓:𝑂–1-1→𝐴)
59	57, 58	nsyl 141	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝑓:𝑂–1-1-onto→𝐴)
60	59	nexdv 1966	. . 3 ⊢ (𝐴 ∈ 𝑉 → ¬ ∃𝑓 𝑓:𝑂–1-1-onto→𝐴)
61		ensym 9001	. . . 4 ⊢ (𝐴 ≈ 𝑂 → 𝑂 ≈ 𝐴)
62		bren 8954	. . . 4 ⊢ (𝑂 ≈ 𝐴 ↔ ∃𝑓 𝑓:𝑂–1-1-onto→𝐴)
63	61, 62	sylib 221	. . 3 ⊢ (𝐴 ≈ 𝑂 → ∃𝑓 𝑓:𝑂–1-1-onto→𝐴)
64	60, 63	nsyl 141	. 2 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝐴 ≈ 𝑂)
65		brsdom 8972	. 2 ⊢ (𝐴 ≺ 𝑂 ↔ (𝐴 ≼ 𝑂 ∧ ¬ 𝐴 ≈ 𝑂))
66	52, 64, 65	sylanbrc 594	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≺ 𝑂)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570  ∃wex 1809   ∈ wcel 2143  ∀wral 3079  Vcvv 3455  [wsbc 3745   ∩ cin 3905   ⊆ wss 3906  ∅c0 4287  𝒫 cpw 4563  {csn 4590  ⟨cop 4596  ∪ cuni 4873   class class class wbr 5110  {copab 5174   We wwe 5615   × cxp 5661  ^◡ccnv 5662  dom cdm 5663   “ cima 5666  Rel wrel 5668  –1-1→wf1 6535  –1-1-onto→wf1o 6537  ‘cfv 6538  (class class class)co 7412   ≈ cen 8941   ≼ cdom 8942   ≺ csdm 8943

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7369  df-ov 7415  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-er 8695  df-en 8945  df-dom 8946  df-sdom 8947  df-oi 9473

This theorem is referenced by: (None)