Theorem canthwe 10606

Description: The set of well-orders of a set 𝐴 strictly dominates 𝐴. A stronger form of canth2 9098. 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 1148	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑥 ⊆ 𝐴)
2		velpw 4559	. . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
3	1, 2	sylibr 236	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑥 ∈ 𝒫 𝐴)
4		simp2 1149	. . . . . . . . 9 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑟 ⊆ (𝑥 × 𝑥))
5		xpss12 5660	. . . . . . . . . 10 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ⊆ 𝐴) → (𝑥 × 𝑥) ⊆ (𝐴 × 𝐴))
6	1, 1, 5	syl2anc 593	. . . . . . . . 9 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → (𝑥 × 𝑥) ⊆ (𝐴 × 𝐴))
7	4, 6	sstrd 3946	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑟 ⊆ (𝐴 × 𝐴))
8		velpw 4559	. . . . . . . 8 ⊢ (𝑟 ∈ 𝒫 (𝐴 × 𝐴) ↔ 𝑟 ⊆ (𝐴 × 𝐴))
9	7, 8	sylibr 236	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → 𝑟 ∈ 𝒫 (𝐴 × 𝐴))
10	3, 9	jca 519	. . . . . 6 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) → (𝑥 ∈ 𝒫 𝐴 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐴)))
11	10	ssopab2i 5519	. . . . 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 5651	. . . . 5 ⊢ (𝒫 𝐴 × 𝒫 (𝐴 × 𝐴)) = {⟨𝑥, 𝑟⟩ ∣ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐴))}
14	11, 12, 13	3sstr4i 3987	. . . 4 ⊢ 𝑂 ⊆ (𝒫 𝐴 × 𝒫 (𝐴 × 𝐴))
15		pwexg 5334	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
16		sqxpexg 7734	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × 𝐴) ∈ V)
17	16	pwexd 5335	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝒫 (𝐴 × 𝐴) ∈ V)
18	15, 17	xpexd 7730	. . . 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 596	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝑂 ∈ V)
21		simpr 488	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑢 ∈ 𝐴) → 𝑢 ∈ 𝐴)
22	21	snssd 4744	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑢 ∈ 𝐴) → {𝑢} ⊆ 𝐴)
23		0ss 4353	. . . . . . . 8 ⊢ ∅ ⊆ ({𝑢} × {𝑢})
24	23	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑢 ∈ 𝐴) → ∅ ⊆ ({𝑢} × {𝑢}))
25		rel0 5769	. . . . . . . 8 ⊢ Rel ∅
26		br0 5148	. . . . . . . . 9 ⊢ ¬ 𝑢∅𝑢
27		wesn 5734	. . . . . . . . 9 ⊢ (Rel ∅ → (∅ We {𝑢} ↔ ¬ 𝑢∅𝑢))
28	26, 27	mpbiri 260	. . . . . . . 8 ⊢ (Rel ∅ → ∅ We {𝑢})
29	25, 28	mp1i 13	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑢 ∈ 𝐴) → ∅ We {𝑢})
30		vsnex 5391	. . . . . . . 8 ⊢ {𝑢} ∈ V
31		0ex 5256	. . . . . . . 8 ⊢ ∅ ∈ V
32		simpl 486	. . . . . . . . . 10 ⊢ ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → 𝑥 = {𝑢})
33	32	sseq1d 3967	. . . . . . . . 9 ⊢ ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → (𝑥 ⊆ 𝐴 ↔ {𝑢} ⊆ 𝐴))
34		simpr 488	. . . . . . . . . 10 ⊢ ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → 𝑟 = ∅)
35	32	sqxpeqd 5677	. . . . . . . . . 10 ⊢ ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → (𝑥 × 𝑥) = ({𝑢} × {𝑢}))
36	34, 35	sseq12d 3969	. . . . . . . . 9 ⊢ ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → (𝑟 ⊆ (𝑥 × 𝑥) ↔ ∅ ⊆ ({𝑢} × {𝑢})))
37	34, 32	weeq12d 5634	. . . . . . . . 9 ⊢ ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → (𝑟 We 𝑥 ↔ ∅ We {𝑢}))
38	33, 36, 37	3anbi123d 1456	. . . . . . . 8 ⊢ ((𝑥 = {𝑢} ∧ 𝑟 = ∅) → ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ↔ ({𝑢} ⊆ 𝐴 ∧ ∅ ⊆ ({𝑢} × {𝑢}) ∧ ∅ We {𝑢})))
39	30, 31, 38	opelopaba 5505	. . . . . . 7 ⊢ (⟨{𝑢}, ∅⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)} ↔ ({𝑢} ⊆ 𝐴 ∧ ∅ ⊆ ({𝑢} × {𝑢}) ∧ ∅ We {𝑢}))
40	22, 24, 29, 39	syl3anbrc 1356	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑢 ∈ 𝐴) → ⟨{𝑢}, ∅⟩ ∈ {⟨𝑥, 𝑟⟩ ∣ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)})
41	40, 12	eleqtrrdi 2872	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑢 ∈ 𝐴) → ⟨{𝑢}, ∅⟩ ∈ 𝑂)
42	41	ex 416	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑢 ∈ 𝐴 → ⟨{𝑢}, ∅⟩ ∈ 𝑂))
43		eqid 2761	. . . . . . 7 ⊢ ∅ = ∅
44		vsnex 5391	. . . . . . . 8 ⊢ {𝑣} ∈ V
45	44, 31	opth2 5447	. . . . . . 7 ⊢ (⟨{𝑢}, ∅⟩ = ⟨{𝑣}, ∅⟩ ↔ ({𝑢} = {𝑣} ∧ ∅ = ∅))
46	43, 45	mpbiran2 720	. . . . . 6 ⊢ (⟨{𝑢}, ∅⟩ = ⟨{𝑣}, ∅⟩ ↔ {𝑢} = {𝑣})
47		sneqbg 4800	. . . . . . 7 ⊢ (𝑢 ∈ V → ({𝑢} = {𝑣} ↔ 𝑢 = 𝑣))
48	47	elv 3458	. . . . . 6 ⊢ ({𝑢} = {𝑣} ↔ 𝑢 = 𝑣)
49	46, 48	bitri 277	. . . . 5 ⊢ (⟨{𝑢}, ∅⟩ = ⟨{𝑣}, ∅⟩ ↔ 𝑢 = 𝑣)
50	49	2a1i 12	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) → (⟨{𝑢}, ∅⟩ = ⟨{𝑣}, ∅⟩ ↔ 𝑢 = 𝑣)))
51	42, 50	dom2d 8970	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑂 ∈ V → 𝐴 ≼ 𝑂))
52	20, 51	mpd 15	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≼ 𝑂)
53		eqid 2761	. . . . . . 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 10585	. . . . . 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 2761	. . . . . 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 2761	. . . . . 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 10605	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝑓:𝑂–1-1→𝐴)
58		f1of1 6801	. . . . 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 1955	. . 3 ⊢ (𝐴 ∈ 𝑉 → ¬ ∃𝑓 𝑓:𝑂–1-1-onto→𝐴)
61		ensym 8980	. . . 4 ⊢ (𝐴 ≈ 𝑂 → 𝑂 ≈ 𝐴)
62		bren 8933	. . . 4 ⊢ (𝑂 ≈ 𝐴 ↔ ∃𝑓 𝑓:𝑂–1-1-onto→𝐴)
63	61, 62	sylib 220	. . 3 ⊢ (𝐴 ≈ 𝑂 → ∃𝑓 𝑓:𝑂–1-1-onto→𝐴)
64	60, 63	nsyl 140	. 2 ⊢ (𝐴 ∈ 𝑉 → ¬ 𝐴 ≈ 𝑂)
65		brsdom 8951	. 2 ⊢ (𝐴 ≺ 𝑂 ↔ (𝐴 ≼ 𝑂 ∧ ¬ 𝐴 ≈ 𝑂))
66	52, 64, 65	sylanbrc 592	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≺ 𝑂)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559  ∃wex 1798   ∈ wcel 2141  ∀wral 3075  Vcvv 3453  [wsbc 3744   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  𝒫 cpw 4554  {csn 4581  ⟨cop 4587  ∪ cuni 4864   class class class wbr 5099  {copab 5161   We wwe 5597   × cxp 5643  ^◡ccnv 5644  dom cdm 5645   “ cima 5648  Rel wrel 5650  –1-1→wf1 6514  –1-1-onto→wf1o 6516  ‘cfv 6517  (class class class)co 7392   ≈ cen 8920   ≼ cdom 8921   ≺ csdm 8922

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-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  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-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-se 5599  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-isom 6526  df-riota 7349  df-ov 7395  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-er 8673  df-en 8924  df-dom 8925  df-sdom 8926  df-oi 9455

This theorem is referenced by: (None)