Theorem jltaleph 35977

Description: The value of 𝐽 is less than an aleph if its first two arguments are less than that aleph. Proposition 15.11 of [TakeutiZaring] p. 156. (Contributed by BTernaryTau, 16-Oct-2026.)

Assertion

Ref	Expression
jltaleph	⊢ ((𝐴 ∈ (ℵ‘𝐷) ∧ 𝐵 ∈ (ℵ‘𝐷) ∧ 𝐶 ∈ 9o) → (𝐽‘⟨𝐴, 𝐵, 𝐶⟩) ∈ (ℵ‘𝐷))

Proof of Theorem jltaleph

Dummy variables 𝑛 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		alephon 10155	. . . . . . 7 ⊢ (ℵ‘𝐷) ∈ On
2	1	oneli 6481	. . . . . 6 ⊢ (𝐴 ∈ (ℵ‘𝐷) → 𝐴 ∈ On)
3	2	3ad2ant1 1151	. . . . 5 ⊢ ((𝐴 ∈ (ℵ‘𝐷) ∧ 𝐵 ∈ (ℵ‘𝐷) ∧ 𝐶 ∈ 9o) → 𝐴 ∈ On)
4	1	oneli 6481	. . . . . 6 ⊢ (𝐵 ∈ (ℵ‘𝐷) → 𝐵 ∈ On)
5	4	3ad2ant2 1152	. . . . 5 ⊢ ((𝐴 ∈ (ℵ‘𝐷) ∧ 𝐵 ∈ (ℵ‘𝐷) ∧ 𝐶 ∈ 9o) → 𝐵 ∈ On)
6		simp3 1156	. . . . 5 ⊢ ((𝐴 ∈ (ℵ‘𝐷) ∧ 𝐵 ∈ (ℵ‘𝐷) ∧ 𝐶 ∈ 9o) → 𝐶 ∈ 9o)
7		ovexd 7459	. . . . 5 ⊢ ((𝐴 ∈ (ℵ‘𝐷) ∧ 𝐵 ∈ (ℵ‘𝐷) ∧ 𝐶 ∈ 9o) → ((9o ·o (𝐽0‘⟨𝐴, 𝐵⟩)) +o 𝐶) ∈ V)
8		opeq1 4839	. . . . . . . . 9 ⊢ (𝐴 = 𝑥 → ⟨𝐴, 𝑦⟩ = ⟨𝑥, 𝑦⟩)
9	8	fveq2d 6894	. . . . . . . 8 ⊢ (𝐴 = 𝑥 → (𝐽0‘⟨𝐴, 𝑦⟩) = (𝐽0‘⟨𝑥, 𝑦⟩))
10	9	oveq2d 7440	. . . . . . 7 ⊢ (𝐴 = 𝑥 → (9o ·o (𝐽0‘⟨𝐴, 𝑦⟩)) = (9o ·o (𝐽0‘⟨𝑥, 𝑦⟩)))
11	10	oveq1d 7439	. . . . . 6 ⊢ (𝐴 = 𝑥 → ((9o ·o (𝐽0‘⟨𝐴, 𝑦⟩)) +o 𝑛) = ((9o ·o (𝐽0‘⟨𝑥, 𝑦⟩)) +o 𝑛))
12	11	adantl 487	. . . . 5 ⊢ (((𝐴 ∈ (ℵ‘𝐷) ∧ 𝐵 ∈ (ℵ‘𝐷) ∧ 𝐶 ∈ 9o) ∧ 𝐴 = 𝑥) → ((9o ·o (𝐽0‘⟨𝐴, 𝑦⟩)) +o 𝑛) = ((9o ·o (𝐽0‘⟨𝑥, 𝑦⟩)) +o 𝑛))
13		opeq2 4840	. . . . . . . . 9 ⊢ (𝐵 = 𝑦 → ⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝑦⟩)
14	13	fveq2d 6894	. . . . . . . 8 ⊢ (𝐵 = 𝑦 → (𝐽0‘⟨𝐴, 𝐵⟩) = (𝐽0‘⟨𝐴, 𝑦⟩))
15	14	oveq2d 7440	. . . . . . 7 ⊢ (𝐵 = 𝑦 → (9o ·o (𝐽0‘⟨𝐴, 𝐵⟩)) = (9o ·o (𝐽0‘⟨𝐴, 𝑦⟩)))
16	15	oveq1d 7439	. . . . . 6 ⊢ (𝐵 = 𝑦 → ((9o ·o (𝐽0‘⟨𝐴, 𝐵⟩)) +o 𝑛) = ((9o ·o (𝐽0‘⟨𝐴, 𝑦⟩)) +o 𝑛))
17	16	adantl 487	. . . . 5 ⊢ (((𝐴 ∈ (ℵ‘𝐷) ∧ 𝐵 ∈ (ℵ‘𝐷) ∧ 𝐶 ∈ 9o) ∧ 𝐵 = 𝑦) → ((9o ·o (𝐽0‘⟨𝐴, 𝐵⟩)) +o 𝑛) = ((9o ·o (𝐽0‘⟨𝐴, 𝑦⟩)) +o 𝑛))
18		oveq2 7432	. . . . . 6 ⊢ (𝐶 = 𝑛 → ((9o ·o (𝐽0‘⟨𝐴, 𝐵⟩)) +o 𝐶) = ((9o ·o (𝐽0‘⟨𝐴, 𝐵⟩)) +o 𝑛))
19	18	adantl 487	. . . . 5 ⊢ (((𝐴 ∈ (ℵ‘𝐷) ∧ 𝐵 ∈ (ℵ‘𝐷) ∧ 𝐶 ∈ 9o) ∧ 𝐶 = 𝑛) → ((9o ·o (𝐽0‘⟨𝐴, 𝐵⟩)) +o 𝐶) = ((9o ·o (𝐽0‘⟨𝐴, 𝐵⟩)) +o 𝑛))
20		df-j 35943	. . . . 5 ⊢ 𝐽 = (𝑥 ∈ On, 𝑦 ∈ On, 𝑛 ∈ 9o ↦ ((9o ·o (𝐽0‘⟨𝑥, 𝑦⟩)) +o 𝑛))
21	3, 5, 6, 7, 12, 17, 19, 20	mpt3fvot2d 7694	. . . 4 ⊢ ((𝐴 ∈ (ℵ‘𝐷) ∧ 𝐵 ∈ (ℵ‘𝐷) ∧ 𝐶 ∈ 9o) → (𝐽‘⟨𝐴, 𝐵, 𝐶⟩) = ((9o ·o (𝐽0‘⟨𝐴, 𝐵⟩)) +o 𝐶))
22	21	adantr 486	. . 3 ⊢ (((𝐴 ∈ (ℵ‘𝐷) ∧ 𝐵 ∈ (ℵ‘𝐷) ∧ 𝐶 ∈ 9o) ∧ (𝐽0‘⟨𝐴, 𝐵⟩) ∈ ω) → (𝐽‘⟨𝐴, 𝐵, 𝐶⟩) = ((9o ·o (𝐽0‘⟨𝐴, 𝐵⟩)) +o 𝐶))
23	6	anim1ci 628	. . . 4 ⊢ (((𝐴 ∈ (ℵ‘𝐷) ∧ 𝐵 ∈ (ℵ‘𝐷) ∧ 𝐶 ∈ 9o) ∧ (𝐽0‘⟨𝐴, 𝐵⟩) ∈ ω) → ((𝐽0‘⟨𝐴, 𝐵⟩) ∈ ω ∧ 𝐶 ∈ 9o))
24		alephfnon 10151	. . . . . . . . . . 11 ⊢ ℵ Fn On
25	24	fndmi 6648	. . . . . . . . . 10 ⊢ dom ℵ = On
26	25	eleq2i 2852	. . . . . . . . 9 ⊢ (𝐷 ∈ dom ℵ ↔ 𝐷 ∈ On)
27		ndmfv 6922	. . . . . . . . 9 ⊢ (¬ 𝐷 ∈ dom ℵ → (ℵ‘𝐷) = ∅)
28	26, 27	sylnbir 334	. . . . . . . 8 ⊢ (¬ 𝐷 ∈ On → (ℵ‘𝐷) = ∅)
29		nel02 4284	. . . . . . . 8 ⊢ ((ℵ‘𝐷) = ∅ → ¬ 𝐴 ∈ (ℵ‘𝐷))
30	28, 29	syl 18	. . . . . . 7 ⊢ (¬ 𝐷 ∈ On → ¬ 𝐴 ∈ (ℵ‘𝐷))
31	30	con4i 115	. . . . . 6 ⊢ (𝐴 ∈ (ℵ‘𝐷) → 𝐷 ∈ On)
32	31	3ad2ant1 1151	. . . . 5 ⊢ ((𝐴 ∈ (ℵ‘𝐷) ∧ 𝐵 ∈ (ℵ‘𝐷) ∧ 𝐶 ∈ 9o) → 𝐷 ∈ On)
33	32	adantr 486	. . . 4 ⊢ (((𝐴 ∈ (ℵ‘𝐷) ∧ 𝐵 ∈ (ℵ‘𝐷) ∧ 𝐶 ∈ 9o) ∧ (𝐽0‘⟨𝐴, 𝐵⟩) ∈ ω) → 𝐷 ∈ On)
34		alephgeom 10168	. . . . . 6 ⊢ (𝐷 ∈ On ↔ ω ⊆ (ℵ‘𝐷))
35	34	bilani 510	. . . . 5 ⊢ ((((𝐽0‘⟨𝐴, 𝐵⟩) ∈ ω ∧ 𝐶 ∈ 9o) ∧ 𝐷 ∈ On) → ω ⊆ (ℵ‘𝐷))
36		9onn 35848	. . . . . . . 8 ⊢ 9o ∈ ω
37		nnmcl 8629	. . . . . . . 8 ⊢ ((9o ∈ ω ∧ (𝐽0‘⟨𝐴, 𝐵⟩) ∈ ω) → (9o ·o (𝐽0‘⟨𝐴, 𝐵⟩)) ∈ ω)
38	36, 37	mpan 703	. . . . . . 7 ⊢ ((𝐽0‘⟨𝐴, 𝐵⟩) ∈ ω → (9o ·o (𝐽0‘⟨𝐴, 𝐵⟩)) ∈ ω)
39		elnn 7892	. . . . . . . 8 ⊢ ((𝐶 ∈ 9o ∧ 9o ∈ ω) → 𝐶 ∈ ω)
40	36, 39	mpan2 704	. . . . . . 7 ⊢ (𝐶 ∈ 9o → 𝐶 ∈ ω)
41		nnacl 8628	. . . . . . 7 ⊢ (((9o ·o (𝐽0‘⟨𝐴, 𝐵⟩)) ∈ ω ∧ 𝐶 ∈ ω) → ((9o ·o (𝐽0‘⟨𝐴, 𝐵⟩)) +o 𝐶) ∈ ω)
42	38, 40, 41	syl2an 608	. . . . . 6 ⊢ (((𝐽0‘⟨𝐴, 𝐵⟩) ∈ ω ∧ 𝐶 ∈ 9o) → ((9o ·o (𝐽0‘⟨𝐴, 𝐵⟩)) +o 𝐶) ∈ ω)
43	42	adantr 486	. . . . 5 ⊢ ((((𝐽0‘⟨𝐴, 𝐵⟩) ∈ ω ∧ 𝐶 ∈ 9o) ∧ 𝐷 ∈ On) → ((9o ·o (𝐽0‘⟨𝐴, 𝐵⟩)) +o 𝐶) ∈ ω)
44	35, 43	sseldd 3910	. . . 4 ⊢ ((((𝐽0‘⟨𝐴, 𝐵⟩) ∈ ω ∧ 𝐶 ∈ 9o) ∧ 𝐷 ∈ On) → ((9o ·o (𝐽0‘⟨𝐴, 𝐵⟩)) +o 𝐶) ∈ (ℵ‘𝐷))
45	23, 33, 44	syl2anc 596	. . 3 ⊢ (((𝐴 ∈ (ℵ‘𝐷) ∧ 𝐵 ∈ (ℵ‘𝐷) ∧ 𝐶 ∈ 9o) ∧ (𝐽0‘⟨𝐴, 𝐵⟩) ∈ ω) → ((9o ·o (𝐽0‘⟨𝐴, 𝐵⟩)) +o 𝐶) ∈ (ℵ‘𝐷))
46	22, 45	eqeltrd 2860	. 2 ⊢ (((𝐴 ∈ (ℵ‘𝐷) ∧ 𝐵 ∈ (ℵ‘𝐷) ∧ 𝐶 ∈ 9o) ∧ (𝐽0‘⟨𝐴, 𝐵⟩) ∈ ω) → (𝐽‘⟨𝐴, 𝐵, 𝐶⟩) ∈ (ℵ‘𝐷))
47	21	adantr 486	. . . . 5 ⊢ (((𝐴 ∈ (ℵ‘𝐷) ∧ 𝐵 ∈ (ℵ‘𝐷) ∧ 𝐶 ∈ 9o) ∧ ¬ (𝐽0‘⟨𝐴, 𝐵⟩) ∈ ω) → (𝐽‘⟨𝐴, 𝐵, 𝐶⟩) = ((9o ·o (𝐽0‘⟨𝐴, 𝐵⟩)) +o 𝐶))
48		j0iso 35955	. . . . . . . . . 10 ⊢ 𝐽0 Isom 𝑅0, E ((On × On), On)
49		isof1o 7335	. . . . . . . . . 10 ⊢ (𝐽0 Isom 𝑅0, E ((On × On), On) → 𝐽0:(On × On)–1-1-onto→On)
50		f1of 6829	. . . . . . . . . 10 ⊢ (𝐽0:(On × On)–1-1-onto→On → 𝐽0:(On × On)⟶On)
51	48, 49, 50	mp2b 10	. . . . . . . . 9 ⊢ 𝐽0:(On × On)⟶On
52		0elon 6421	. . . . . . . . 9 ⊢ ∅ ∈ On
53	51, 52	f0cli 7103	. . . . . . . 8 ⊢ (𝐽0‘⟨𝐴, 𝐵⟩) ∈ On
54		onfin 9238	. . . . . . . . . 10 ⊢ ((𝐽0‘⟨𝐴, 𝐵⟩) ∈ On → ((𝐽0‘⟨𝐴, 𝐵⟩) ∈ Fin ↔ (𝐽0‘⟨𝐴, 𝐵⟩) ∈ ω))
55	54	notbid 321	. . . . . . . . 9 ⊢ ((𝐽0‘⟨𝐴, 𝐵⟩) ∈ On → (¬ (𝐽0‘⟨𝐴, 𝐵⟩) ∈ Fin ↔ ¬ (𝐽0‘⟨𝐴, 𝐵⟩) ∈ ω))
56		onenon 10037	. . . . . . . . . 10 ⊢ ((𝐽0‘⟨𝐴, 𝐵⟩) ∈ On → (𝐽0‘⟨𝐴, 𝐵⟩) ∈ dom card)
57		infinfnum 35825	. . . . . . . . . 10 ⊢ ((𝐽0‘⟨𝐴, 𝐵⟩) ∈ dom card → (¬ (𝐽0‘⟨𝐴, 𝐵⟩) ∈ Fin ↔ ω ≼ (𝐽0‘⟨𝐴, 𝐵⟩)))
58	56, 57	syl 18	. . . . . . . . 9 ⊢ ((𝐽0‘⟨𝐴, 𝐵⟩) ∈ On → (¬ (𝐽0‘⟨𝐴, 𝐵⟩) ∈ Fin ↔ ω ≼ (𝐽0‘⟨𝐴, 𝐵⟩)))
59	55, 58	bitr3d 284	. . . . . . . 8 ⊢ ((𝐽0‘⟨𝐴, 𝐵⟩) ∈ On → (¬ (𝐽0‘⟨𝐴, 𝐵⟩) ∈ ω ↔ ω ≼ (𝐽0‘⟨𝐴, 𝐵⟩)))
60	53, 59	ax-mp 5	. . . . . . 7 ⊢ (¬ (𝐽0‘⟨𝐴, 𝐵⟩) ∈ ω ↔ ω ≼ (𝐽0‘⟨𝐴, 𝐵⟩))
61		jltalephlem 35976	. . . . . . 7 ⊢ ((𝐶 ∈ 9o ∧ ω ≼ (𝐽0‘⟨𝐴, 𝐵⟩)) → ((9o ·o (𝐽0‘⟨𝐴, 𝐵⟩)) +o 𝐶) ≈ (𝐽0‘⟨𝐴, 𝐵⟩))
62	60, 61	sylan2b 606	. . . . . 6 ⊢ ((𝐶 ∈ 9o ∧ ¬ (𝐽0‘⟨𝐴, 𝐵⟩) ∈ ω) → ((9o ·o (𝐽0‘⟨𝐴, 𝐵⟩)) +o 𝐶) ≈ (𝐽0‘⟨𝐴, 𝐵⟩))
63	62	3ad2antl3 1206	. . . . 5 ⊢ (((𝐴 ∈ (ℵ‘𝐷) ∧ 𝐵 ∈ (ℵ‘𝐷) ∧ 𝐶 ∈ 9o) ∧ ¬ (𝐽0‘⟨𝐴, 𝐵⟩) ∈ ω) → ((9o ·o (𝐽0‘⟨𝐴, 𝐵⟩)) +o 𝐶) ≈ (𝐽0‘⟨𝐴, 𝐵⟩))
64	47, 63	eqbrtrd 5126	. . . 4 ⊢ (((𝐴 ∈ (ℵ‘𝐷) ∧ 𝐵 ∈ (ℵ‘𝐷) ∧ 𝐶 ∈ 9o) ∧ ¬ (𝐽0‘⟨𝐴, 𝐵⟩) ∈ ω) → (𝐽‘⟨𝐴, 𝐵, 𝐶⟩) ≈ (𝐽0‘⟨𝐴, 𝐵⟩))
65		j0ltaleph 35959	. . . . . . 7 ⊢ ((𝐴 ∈ (ℵ‘𝐷) ∧ 𝐵 ∈ (ℵ‘𝐷)) → (𝐽0‘⟨𝐴, 𝐵⟩) ∈ (ℵ‘𝐷))
66		alephsdom 10172	. . . . . . . . 9 ⊢ (((𝐽0‘⟨𝐴, 𝐵⟩) ∈ On ∧ 𝐷 ∈ On) → ((𝐽0‘⟨𝐴, 𝐵⟩) ∈ (ℵ‘𝐷) ↔ (𝐽0‘⟨𝐴, 𝐵⟩) ≺ (ℵ‘𝐷)))
67	53, 31, 66	sylancr 599	. . . . . . . 8 ⊢ (𝐴 ∈ (ℵ‘𝐷) → ((𝐽0‘⟨𝐴, 𝐵⟩) ∈ (ℵ‘𝐷) ↔ (𝐽0‘⟨𝐴, 𝐵⟩) ≺ (ℵ‘𝐷)))
68	67	adantr 486	. . . . . . 7 ⊢ ((𝐴 ∈ (ℵ‘𝐷) ∧ 𝐵 ∈ (ℵ‘𝐷)) → ((𝐽0‘⟨𝐴, 𝐵⟩) ∈ (ℵ‘𝐷) ↔ (𝐽0‘⟨𝐴, 𝐵⟩) ≺ (ℵ‘𝐷)))
69	65, 68	mpbid 235	. . . . . 6 ⊢ ((𝐴 ∈ (ℵ‘𝐷) ∧ 𝐵 ∈ (ℵ‘𝐷)) → (𝐽0‘⟨𝐴, 𝐵⟩) ≺ (ℵ‘𝐷))
70	69	3adant3 1150	. . . . 5 ⊢ ((𝐴 ∈ (ℵ‘𝐷) ∧ 𝐵 ∈ (ℵ‘𝐷) ∧ 𝐶 ∈ 9o) → (𝐽0‘⟨𝐴, 𝐵⟩) ≺ (ℵ‘𝐷))
71	70	adantr 486	. . . 4 ⊢ (((𝐴 ∈ (ℵ‘𝐷) ∧ 𝐵 ∈ (ℵ‘𝐷) ∧ 𝐶 ∈ 9o) ∧ ¬ (𝐽0‘⟨𝐴, 𝐵⟩) ∈ ω) → (𝐽0‘⟨𝐴, 𝐵⟩) ≺ (ℵ‘𝐷))
72		ensdomtr 9140	. . . 4 ⊢ (((𝐽‘⟨𝐴, 𝐵, 𝐶⟩) ≈ (𝐽0‘⟨𝐴, 𝐵⟩) ∧ (𝐽0‘⟨𝐴, 𝐵⟩) ≺ (ℵ‘𝐷)) → (𝐽‘⟨𝐴, 𝐵, 𝐶⟩) ≺ (ℵ‘𝐷))
73	64, 71, 72	syl2anc 596	. . 3 ⊢ (((𝐴 ∈ (ℵ‘𝐷) ∧ 𝐵 ∈ (ℵ‘𝐷) ∧ 𝐶 ∈ 9o) ∧ ¬ (𝐽0‘⟨𝐴, 𝐵⟩) ∈ ω) → (𝐽‘⟨𝐴, 𝐵, 𝐶⟩) ≺ (ℵ‘𝐷))
74		jf1 35961	. . . . . . . 8 ⊢ 𝐽:((On × On) × 9o)–1-1→On
75		f1f 6783	. . . . . . . 8 ⊢ (𝐽:((On × On) × 9o)–1-1→On → 𝐽:((On × On) × 9o)⟶On)
76	74, 75	ax-mp 5	. . . . . . 7 ⊢ 𝐽:((On × On) × 9o)⟶On
77	76, 52	f0cli 7103	. . . . . 6 ⊢ (𝐽‘⟨𝐴, 𝐵, 𝐶⟩) ∈ On
78		alephsdom 10172	. . . . . 6 ⊢ (((𝐽‘⟨𝐴, 𝐵, 𝐶⟩) ∈ On ∧ 𝐷 ∈ On) → ((𝐽‘⟨𝐴, 𝐵, 𝐶⟩) ∈ (ℵ‘𝐷) ↔ (𝐽‘⟨𝐴, 𝐵, 𝐶⟩) ≺ (ℵ‘𝐷)))
79	77, 31, 78	sylancr 599	. . . . 5 ⊢ (𝐴 ∈ (ℵ‘𝐷) → ((𝐽‘⟨𝐴, 𝐵, 𝐶⟩) ∈ (ℵ‘𝐷) ↔ (𝐽‘⟨𝐴, 𝐵, 𝐶⟩) ≺ (ℵ‘𝐷)))
80	79	3ad2ant1 1151	. . . 4 ⊢ ((𝐴 ∈ (ℵ‘𝐷) ∧ 𝐵 ∈ (ℵ‘𝐷) ∧ 𝐶 ∈ 9o) → ((𝐽‘⟨𝐴, 𝐵, 𝐶⟩) ∈ (ℵ‘𝐷) ↔ (𝐽‘⟨𝐴, 𝐵, 𝐶⟩) ≺ (ℵ‘𝐷)))
81	80	adantr 486	. . 3 ⊢ (((𝐴 ∈ (ℵ‘𝐷) ∧ 𝐵 ∈ (ℵ‘𝐷) ∧ 𝐶 ∈ 9o) ∧ ¬ (𝐽0‘⟨𝐴, 𝐵⟩) ∈ ω) → ((𝐽‘⟨𝐴, 𝐵, 𝐶⟩) ∈ (ℵ‘𝐷) ↔ (𝐽‘⟨𝐴, 𝐵, 𝐶⟩) ≺ (ℵ‘𝐷)))
82	73, 81	mpbird 260	. 2 ⊢ (((𝐴 ∈ (ℵ‘𝐷) ∧ 𝐵 ∈ (ℵ‘𝐷) ∧ 𝐶 ∈ 9o) ∧ ¬ (𝐽0‘⟨𝐴, 𝐵⟩) ∈ ω) → (𝐽‘⟨𝐴, 𝐵, 𝐶⟩) ∈ (ℵ‘𝐷))
83	46, 82	pm2.61dan 825	1 ⊢ ((𝐴 ∈ (ℵ‘𝐷) ∧ 𝐵 ∈ (ℵ‘𝐷) ∧ 𝐶 ∈ 9o) → (𝐽‘⟨𝐴, 𝐵, 𝐶⟩) ∈ (ℵ‘𝐷))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  Vcvv 3450   ⊆ wss 3896  ∅c0 4277  ⟨cop 4834  ⟨cotp 4857   class class class wbr 5102   E cep 5550   × cxp 5649  dom cdm 5651  Oncon0 6365  ⟶wf 6539  –1-1→wf1 6540  –1-1-onto→wf1o 6542  ‘cfv 6543   Isom wiso 6544  (class class class)co 7424  ωcom 7881   +_o coa 8481   ·_o comu 8482   ≈ cen 8978   ≼ cdom 8979   ≺ csdm 8980  Fincfn 8981  cardccrd 10023  ℵcale 10024  9_oc9o 35833  𝑅₀cr0 35932  𝐽₀ccj0 35933  𝐽cj 35934

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755  ax-inf2 9649

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-ot 4858  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-mpt3 7688  df-rpss 7743  df-om 7882  df-1st 8005  df-2nd 8006  df-frecs 8307  df-wrecs 8338  df-recs 8387  df-rdg 8426  df-seqom 8466  df-1o 8484  df-2o 8485  df-3o 8486  df-4o 8487  df-oadd 8488  df-omul 8489  df-er 8725  df-map 8857  df-en 8982  df-dom 8983  df-sdom 8984  df-fin 8985  df-oi 9511  df-har 9558  df-wdom 9566  df-dju 9989  df-card 10027  df-aleph 10028  df-fin2 10371  df-fin4 10372  df-fin3 10373  df-fin5 10374  df-fin6 10375  df-fin7 10376  df-5o 35834  df-6o 35835  df-7o 35836  df-8o 35837  df-9o 35838  df-lexo 35940  df-r0 35941  df-j0 35942  df-j 35943

This theorem is used by: (None)