Theorem canthwe 10604

Description: The set of well-orders of a set 𝐴 strictly dominates 𝐴. A stronger form of canth2 9094. 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 1136	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑥 ⊆ 𝐴)
2		velpw 4568	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
3	1, 2	sylibr 234	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑥 ∈ 𝒫 𝐴)
4		simp2 1137	. . . . . . . . 9 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑟 ⊆ (𝑥 × 𝑥))
5		xpss12 5653	. . . . . . . . . 10 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐴) → (𝑥 × 𝑥) ⊆ (𝐴 × 𝐴))
6	1, 1, 5	syl2anc 584	. . . . . . . . 9 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → (𝑥 × 𝑥) ⊆ (𝐴 × 𝐴))
7	4, 6	sstrd 3957	. . . . . . . 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 234	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑟 ∈ 𝒫 (𝐴 × 𝐴))
10	3, 9	jca 511	. . . . . 6 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → (𝑥 ∈ 𝒫 𝐴 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐴)))
11	10	ssopab2i 5510	. . . . 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 5644	. . . . 5 ⊢ (𝒫 𝐴 × 𝒫 (𝐴 × 𝐴)) = {⟨𝑥, 𝑟⟩ ∣ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐴))}
14	11, 12, 13	3sstr4i 3998	. . . 4 ⊢ 𝑂 ⊆ (𝒫 𝐴 × 𝒫 (𝐴 × 𝐴))
15		pwexg 5333	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
16		sqxpexg 7731	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × 𝐴) ∈ V)
17	16	pwexd 5334	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝒫 (𝐴 × 𝐴) ∈ V)
18	15, 17	xpexd 7727	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝒫 𝐴 × 𝒫 (𝐴 × 𝐴)) ∈ V)
19		ssexg 5278	. . . 4 ⊢ ((𝑂 ⊆ (𝒫 𝐴 × 𝒫 (𝐴 × 𝐴)) ∧ (𝒫 𝐴 × 𝒫 (𝐴 × 𝐴)) ∈ V) → 𝑂 ∈ V)
20	14, 18, 19	sylancr 587	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝑂 ∈ V)
21		simpr 484	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑢 ∈ 𝐴) → 𝑢 ∈ 𝐴)
22	21	snssd 4773	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑢 ∈ 𝐴) → {𝑢} ⊆ 𝐴)
23		0ss 4363	. . . . . . . 8 ⊢ ∅ ⊆ ({𝑢} × {𝑢})
24	23	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑢 ∈ 𝐴) → ∅ ⊆ ({𝑢} × {𝑢}))
25		rel0 5762	. . . . . . . 8 ⊢ Rel ∅
26		br0 5156	. . . . . . . . 9 ⊢ ¬ 𝑢∅𝑢
27		wesn 5727	. . . . . . . . 9 ⊢ (Rel ∅ → (∅ We {𝑢} ↔ ¬ 𝑢∅𝑢))
28	26, 27	mpbiri 258	. . . . . . . 8 ⊢ (Rel ∅ → ∅ We {𝑢})
29	25, 28	mp1i 13	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑢 ∈ 𝐴) → ∅ We {𝑢})
30		vsnex 5389	. . . . . . . 8 ⊢ {𝑢} ∈ V
31		0ex 5262	. . . . . . . 8 ⊢ ∅ ∈ V
32		simpl 482	. . . . . . . . . 10 ⊢ ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → 𝑥 = {𝑢})
33	32	sseq1d 3978	. . . . . . . . 9 ⊢ ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → (𝑥 ⊆ 𝐴 ↔ {𝑢} ⊆ 𝐴))
34		simpr 484	. . . . . . . . . 10 ⊢ ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → 𝑟 = ∅)
35	32	sqxpeqd 5670	. . . . . . . . . 10 ⊢ ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → (𝑥 × 𝑥) = ({𝑢} × {𝑢}))
36	34, 35	sseq12d 3980	. . . . . . . . 9 ⊢ ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → (𝑟 ⊆ (𝑥 × 𝑥) ↔ ∅ ⊆ ({𝑢} × {𝑢})))
37	34, 32	weeq12d 5627	. . . . . . . . 9 ⊢ ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → (𝑟 We 𝑥 ↔ ∅ We {𝑢}))
38	33, 36, 37	3anbi123d 1438	. . . . . . . 8 ⊢ ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ↔ ({𝑢} ⊆ 𝐴 ∧ ∅ ⊆ ({𝑢} × {𝑢}) ∧ ∅ We {𝑢})))
39	30, 31, 38	opelopaba 5496	. . . . . . 7 ⊢ (⟨{𝑢}, ∅⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)} ↔ ({𝑢} ⊆ 𝐴 ∧ ∅ ⊆ ({𝑢} × {𝑢}) ∧ ∅ We {𝑢}))
40	22, 24, 29, 39	syl3anbrc 1344	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑢 ∈ 𝐴) → ⟨{𝑢}, ∅⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)})
41	40, 12	eleqtrrdi 2839	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑢 ∈ 𝐴) → ⟨{𝑢}, ∅⟩ ∈ 𝑂)
42	41	ex 412	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑢 ∈ 𝐴 → ⟨{𝑢}, ∅⟩ ∈ 𝑂))
43		eqid 2729	. . . . . . 7 ⊢ ∅ = ∅
44		vsnex 5389	. . . . . . . 8 ⊢ {𝑣} ∈ V
45	44, 31	opth2 5440	. . . . . . 7 ⊢ (⟨{𝑢}, ∅⟩ = ⟨{𝑣}, ∅⟩ ↔ ({𝑢} = {𝑣} ∧ ∅ = ∅))
46	43, 45	mpbiran2 710	. . . . . 6 ⊢ (⟨{𝑢}, ∅⟩ = ⟨{𝑣}, ∅⟩ ↔ {𝑢} = {𝑣})
47		sneqbg 4807	. . . . . . 7 ⊢ (𝑢 ∈ V → ({𝑢} = {𝑣} ↔ 𝑢 = 𝑣))
48	47	elv 3452	. . . . . 6 ⊢ ({𝑢} = {𝑣} ↔ 𝑢 = 𝑣)
49	46, 48	bitri 275	. . . . 5 ⊢ (⟨{𝑢}, ∅⟩ = ⟨{𝑣}, ∅⟩ ↔ 𝑢 = 𝑣)
50	49	2a1i 12	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (⟨{𝑢}, ∅⟩ = ⟨{𝑣}, ∅⟩ ↔ 𝑢 = 𝑣)))
51	42, 50	dom2d 8964	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑂 ∈ V → 𝐴 ≼ 𝑂))
52	20, 51	mpd 15	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≼ 𝑂)
53		eqid 2729	. . . . . . 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 10583	. . . . . 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 2729	. . . . . 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 2729	. . . . . 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 10603	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝑓:𝑂–1-1→𝐴)
58		f1of1 6799	. . . . 5 ⊢ (𝑓:𝑂–1-1-onto→𝐴 → 𝑓:𝑂–1-1→𝐴)
59	57, 58	nsyl 140	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝑓:𝑂–1-1-onto→𝐴)
60	59	nexdv 1936	. . 3 ⊢ (𝐴 ∈ 𝑉 → ¬ ∃𝑓 𝑓:𝑂–1-1-onto→𝐴)
61		ensym 8974	. . . 4 ⊢ (𝐴 ≈ 𝑂 → 𝑂 ≈ 𝐴)
62		bren 8928	. . . 4 ⊢ (𝑂 ≈ 𝐴 ↔ ∃𝑓 𝑓:𝑂–1-1-onto→𝐴)
63	61, 62	sylib 218	. . 3 ⊢ (𝐴 ≈ 𝑂 → ∃𝑓 𝑓:𝑂–1-1-onto→𝐴)
64	60, 63	nsyl 140	. 2 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝐴 ≈ 𝑂)
65		brsdom 8946	. 2 ⊢ (𝐴 ≺ 𝑂 ↔ (𝐴 ≼ 𝑂 ∧ ¬ 𝐴 ≈ 𝑂))
66	52, 64, 65	sylanbrc 583	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≺ 𝑂)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3044  Vcvv 3447  [wsbc 3753   ∩ cin 3913   ⊆ wss 3914  ∅c0 4296  𝒫 cpw 4563  {csn 4589  ⟨cop 4595  ∪ cuni 4871   class class class wbr 5107  {copab 5169   We wwe 5590   × cxp 5636  ^◡ccnv 5637  dom cdm 5638   “ cima 5641  Rel wrel 5643  –1-1→wf1 6508  –1-1-onto→wf1o 6510  ‘cfv 6511  (class class class)co 7387   ≈ cen 8915   ≼ cdom 8916   ≺ csdm 8917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-oi 9463

This theorem is referenced by: (None)