Theorem jltaleph 36067

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 10187	. . . . . . 7 ⊢ (ℵ‘𝐷) ∈ On
2	1	oneli 6504	. . . . . 6 ⊢ (𝐴 ∈ (ℵ‘𝐷) → 𝐴 ∈ On)
3	2	3ad2ant1 1151	. . . . 5 ⊢ ((𝐴 ∈ (ℵ‘𝐷) ∧ 𝐵 ∈ (ℵ‘𝐷) ∧ 𝐶 ∈ 9o) → 𝐴 ∈ On)
4	1	oneli 6504	. . . . . 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 7486	. . . . 5 ⊢ ((𝐴 ∈ (ℵ‘𝐷) ∧ 𝐵 ∈ (ℵ‘𝐷) ∧ 𝐶 ∈ 9o) → ((9o ·o (𝐽0‘⟨𝐴, 𝐵⟩)) +o 𝐶) ∈ V)
8		opeq1 4845	. . . . . . . . 9 ⊢ (𝐴 = 𝑥 → ⟨𝐴, 𝑦⟩ = ⟨𝑥, 𝑦⟩)
9	8	fveq2d 6920	. . . . . . . 8 ⊢ (𝐴 = 𝑥 → (𝐽0‘⟨𝐴, 𝑦⟩) = (𝐽0‘⟨𝑥, 𝑦⟩))
10	9	oveq2d 7467	. . . . . . 7 ⊢ (𝐴 = 𝑥 → (9o ·o (𝐽0‘⟨𝐴, 𝑦⟩)) = (9o ·o (𝐽0‘⟨𝑥, 𝑦⟩)))
11	10	oveq1d 7466	. . . . . 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 4846	. . . . . . . . 9 ⊢ (𝐵 = 𝑦 → ⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝑦⟩)
14	13	fveq2d 6920	. . . . . . . 8 ⊢ (𝐵 = 𝑦 → (𝐽0‘⟨𝐴, 𝐵⟩) = (𝐽0‘⟨𝐴, 𝑦⟩))
15	14	oveq2d 7467	. . . . . . 7 ⊢ (𝐵 = 𝑦 → (9o ·o (𝐽0‘⟨𝐴, 𝐵⟩)) = (9o ·o (𝐽0‘⟨𝐴, 𝑦⟩)))
16	15	oveq1d 7466	. . . . . 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 7459	. . . . . 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 36033	. . . . 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 7722	. . . 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 10183	. . . . . . . . . . 11 ⊢ ℵ Fn On
25	24	fndmi 6673	. . . . . . . . . 10 ⊢ dom ℵ = On
26	25	eleq2i 2856	. . . . . . . . 9 ⊢ (𝐷 ∈ dom ℵ ↔ 𝐷 ∈ On)
27		ndmfv 6948	. . . . . . . . 9 ⊢ (¬ 𝐷 ∈ dom ℵ → (ℵ‘𝐷) = ∅)
28	26, 27	sylnbir 334	. . . . . . . 8 ⊢ (¬ 𝐷 ∈ On → (ℵ‘𝐷) = ∅)
29		nel02 4288	. . . . . . . 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 10200	. . . . . 6 ⊢ (𝐷 ∈ On ↔ ω ⊆ (ℵ‘𝐷))
35	34	bilani 510	. . . . 5 ⊢ ((((𝐽0‘⟨𝐴, 𝐵⟩) ∈ ω ∧ 𝐶 ∈ 9o) ∧ 𝐷 ∈ On) → ω ⊆ (ℵ‘𝐷))
36		9onn 35913	. . . . . . . 8 ⊢ 9o ∈ ω
37		nnmcl 8659	. . . . . . . 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 7922	. . . . . . . 8 ⊢ ((𝐶 ∈ 9o ∧ 9o ∈ ω) → 𝐶 ∈ ω)
40	36, 39	mpan2 704	. . . . . . 7 ⊢ (𝐶 ∈ 9o → 𝐶 ∈ ω)
41		nnacl 8658	. . . . . . 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 3914	. . . 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 2864	. 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 36045	. . . . . . . . . 10 ⊢ 𝐽0 Isom 𝑅0, E ((On × On), On)
49		isof1o 7362	. . . . . . . . . 10 ⊢ (𝐽0 Isom 𝑅0, E ((On × On), On) → 𝐽0:(On × On)⟶1-1ontoOn)
50		f1of 6854	. . . . . . . . . 10 ⊢ (𝐽0:(On × On)⟶1-1ontoOn → 𝐽0:(On × On)⟶On)
51	48, 49, 50	mp2b 10	. . . . . . . . 9 ⊢ 𝐽0:(On × On)⟶On
52		0elon 6444	. . . . . . . . 9 ⊢ ∅ ∈ On
53	51, 52	f0cli 7130	. . . . . . . 8 ⊢ (𝐽0‘⟨𝐴, 𝐵⟩) ∈ On
54		onfin 9268	. . . . . . . . . 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 10067	. . . . . . . . . 10 ⊢ ((𝐽0‘⟨𝐴, 𝐵⟩) ∈ On → (𝐽0‘⟨𝐴, 𝐵⟩) ∈ dom card)
57		infinfnum 35890	. . . . . . . . . 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 36066	. . . . . . 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 5133	. . . 4 ⊢ (((𝐴 ∈ (ℵ‘𝐷) ∧ 𝐵 ∈ (ℵ‘𝐷) ∧ 𝐶 ∈ 9o) ∧ ¬ (𝐽0‘⟨𝐴, 𝐵⟩) ∈ ω) → (𝐽‘⟨𝐴, 𝐵, 𝐶⟩) ≈ (𝐽0‘⟨𝐴, 𝐵⟩))
65		j0ltaleph 36049	. . . . . . 7 ⊢ ((𝐴 ∈ (ℵ‘𝐷) ∧ 𝐵 ∈ (ℵ‘𝐷)) → (𝐽0‘⟨𝐴, 𝐵⟩) ∈ (ℵ‘𝐷))
66		alephsdom 10204	. . . . . . . . 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 9170	. . . 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 36051	. . . . . . . 8 ⊢ 𝐽:((On × On) × 9o)⟶1-1On
75		f1f 6808	. . . . . . . 8 ⊢ (𝐽:((On × On) × 9o)⟶1-1On → 𝐽:((On × On) × 9o)⟶On)
76	74, 75	ax-mp 5	. . . . . . 7 ⊢ 𝐽:((On × On) × 9o)⟶On
77	76, 52	f0cli 7130	. . . . . 6 ⊢ (𝐽‘⟨𝐴, 𝐵, 𝐶⟩) ∈ On
78		alephsdom 10204	. . . . . 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 1570   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900  ∅c0 4281  ⟨cop 4840  ⟨cotp 4863   class class class wbr 5108   E cep 5557   × cxp 5656  dom cdm 5658  Oncon0 6388  ⟶wf 6562  ⟶1-1wf1 6563  ⟶1-1ontowf1o 6565  ‘cfv 6566   Isom wiso 6567  (class class class)co 7451  ωcom 7911   +_o coa 8511   ·_o comu 8512   ≈ cen 9008   ≼ cdom 9009   ≺ csdm 9010  Fincfn 9011  cardccrd 10053  ℵcale 10054  9_oc9o 35898  𝑅₀cr0 36022  𝐽₀ccj0 36023  𝐽cj 36024

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-inf2 9679

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-ot 4864  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-isom 6575  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-mpt3 7716  df-rpss 7771  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-seqom 8496  df-1o 8514  df-2o 8515  df-3o 8516  df-4o 8517  df-oadd 8518  df-omul 8519  df-er 8755  df-map 8887  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-oi 9541  df-har 9588  df-wdom 9596  df-dju 10019  df-card 10057  df-aleph 10058  df-fin2 10403  df-fin4 10404  df-fin3 10405  df-fin5 10406  df-fin6 10407  df-fin7 10408  df-5o 35899  df-6o 35900  df-7o 35901  df-8o 35902  df-9o 35903  df-lexo 36030  df-r0 36031  df-j0 36032  df-j 36033

This theorem is used by: (None)