Theorem j0ltaleph 35940

Description: The value of 𝐽₀ is less than an aleph if its two arguments are less than that aleph. Proposition 15.1(2) of [TakeutiZaring] p. 154. (Contributed by BTernaryTau, 25-Sep-2026.)

Assertion

Ref	Expression
j0ltaleph	⊢ ((𝐴 ∈ (ℵ‘𝐶) ∧ 𝐵 ∈ (ℵ‘𝐶)) → (𝐽0‘⟨𝐴, 𝐵⟩) ∈ (ℵ‘𝐶))

Proof of Theorem j0ltaleph

Step	Hyp	Ref	Expression
1		omelon 9648	. . . . . . . . 9 ⊢ ω ∈ On
2		j0iso 35936	. . . . . . . . . . 11 ⊢ 𝐽0 Isom 𝑅0, E ((On × On), On)
3		isof1o 7331	. . . . . . . . . . 11 ⊢ (𝐽0 Isom 𝑅0, E ((On × On), On) → 𝐽0:(On × On)–1-1-onto→On)
4		f1of 6824	. . . . . . . . . . 11 ⊢ (𝐽0:(On × On)–1-1-onto→On → 𝐽0:(On × On)⟶On)
5	2, 3, 4	mp2b 10	. . . . . . . . . 10 ⊢ 𝐽0:(On × On)⟶On
6		0elon 6418	. . . . . . . . . 10 ⊢ ∅ ∈ On
7	5, 6	f0cli 7098	. . . . . . . . 9 ⊢ (𝐽0‘⟨𝐴, 𝐵⟩) ∈ On
8		domtriord 9143	. . . . . . . . 9 ⊢ ((ω ∈ On ∧ (𝐽0‘⟨𝐴, 𝐵⟩) ∈ On) → (ω ≼ (𝐽0‘⟨𝐴, 𝐵⟩) ↔ ¬ (𝐽0‘⟨𝐴, 𝐵⟩) ≺ ω))
9	1, 7, 8	mp2an 705	. . . . . . . 8 ⊢ (ω ≼ (𝐽0‘⟨𝐴, 𝐵⟩) ↔ ¬ (𝐽0‘⟨𝐴, 𝐵⟩) ≺ ω)
10	9	con2bii 360	. . . . . . 7 ⊢ ((𝐽0‘⟨𝐴, 𝐵⟩) ≺ ω ↔ ¬ ω ≼ (𝐽0‘⟨𝐴, 𝐵⟩))
11		sdomel 9144	. . . . . . . 8 ⊢ (((𝐽0‘⟨𝐴, 𝐵⟩) ∈ On ∧ ω ∈ On) → ((𝐽0‘⟨𝐴, 𝐵⟩) ≺ ω → (𝐽0‘⟨𝐴, 𝐵⟩) ∈ ω))
12	7, 1, 11	mp2an 705	. . . . . . 7 ⊢ ((𝐽0‘⟨𝐴, 𝐵⟩) ≺ ω → (𝐽0‘⟨𝐴, 𝐵⟩) ∈ ω)
13	10, 12	sylbir 238	. . . . . 6 ⊢ (¬ ω ≼ (𝐽0‘⟨𝐴, 𝐵⟩) → (𝐽0‘⟨𝐴, 𝐵⟩) ∈ ω)
14	13	orri 876	. . . . 5 ⊢ (ω ≼ (𝐽0‘⟨𝐴, 𝐵⟩) ∨ (𝐽0‘⟨𝐴, 𝐵⟩) ∈ ω)
15		alephon 10149	. . . . . . . . . . . 12 ⊢ (ℵ‘𝐶) ∈ On
16	15	oneli 6478	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (ℵ‘𝐶) → 𝐴 ∈ On)
17	15	oneli 6478	. . . . . . . . . . 11 ⊢ (𝐵 ∈ (ℵ‘𝐶) → 𝐵 ∈ On)
18	16, 17	anim12i 625	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (ℵ‘𝐶) ∧ 𝐵 ∈ (ℵ‘𝐶)) → (𝐴 ∈ On ∧ 𝐵 ∈ On))
19	18	3adant3 1150	. . . . . . . . 9 ⊢ ((𝐴 ∈ (ℵ‘𝐶) ∧ 𝐵 ∈ (ℵ‘𝐶) ∧ 𝐶 ∈ On) → (𝐴 ∈ On ∧ 𝐵 ∈ On))
20		j0ltalephlem 35939	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐽0‘⟨𝐴, 𝐵⟩) ≼ (suc (𝐴 ∪ 𝐵) × suc (𝐴 ∪ 𝐵)))
21		xpfi 9312	. . . . . . . . . . . . . . 15 ⊢ ((suc (𝐴 ∪ 𝐵) ∈ Fin ∧ suc (𝐴 ∪ 𝐵) ∈ Fin) → (suc (𝐴 ∪ 𝐵) × suc (𝐴 ∪ 𝐵)) ∈ Fin)
22	21	anidms 577	. . . . . . . . . . . . . 14 ⊢ (suc (𝐴 ∪ 𝐵) ∈ Fin → (suc (𝐴 ∪ 𝐵) × suc (𝐴 ∪ 𝐵)) ∈ Fin)
23		domfi 9205	. . . . . . . . . . . . . 14 ⊢ (((suc (𝐴 ∪ 𝐵) × suc (𝐴 ∪ 𝐵)) ∈ Fin ∧ (𝐽0‘⟨𝐴, 𝐵⟩) ≼ (suc (𝐴 ∪ 𝐵) × suc (𝐴 ∪ 𝐵))) → (𝐽0‘⟨𝐴, 𝐵⟩) ∈ Fin)
24	22, 23	sylan 592	. . . . . . . . . . . . 13 ⊢ ((suc (𝐴 ∪ 𝐵) ∈ Fin ∧ (𝐽0‘⟨𝐴, 𝐵⟩) ≼ (suc (𝐴 ∪ 𝐵) × suc (𝐴 ∪ 𝐵))) → (𝐽0‘⟨𝐴, 𝐵⟩) ∈ Fin)
25	24	expcom 419	. . . . . . . . . . . 12 ⊢ ((𝐽0‘⟨𝐴, 𝐵⟩) ≼ (suc (𝐴 ∪ 𝐵) × suc (𝐴 ∪ 𝐵)) → (suc (𝐴 ∪ 𝐵) ∈ Fin → (𝐽0‘⟨𝐴, 𝐵⟩) ∈ Fin))
26	20, 25	syl 18	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc (𝐴 ∪ 𝐵) ∈ Fin → (𝐽0‘⟨𝐴, 𝐵⟩) ∈ Fin))
27		onenon 10031	. . . . . . . . . . . . 13 ⊢ ((𝐽0‘⟨𝐴, 𝐵⟩) ∈ On → (𝐽0‘⟨𝐴, 𝐵⟩) ∈ dom card)
28		infinfnum 35806	. . . . . . . . . . . . 13 ⊢ ((𝐽0‘⟨𝐴, 𝐵⟩) ∈ dom card → (¬ (𝐽0‘⟨𝐴, 𝐵⟩) ∈ Fin ↔ ω ≼ (𝐽0‘⟨𝐴, 𝐵⟩)))
29	7, 27, 28	mp2b 10	. . . . . . . . . . . 12 ⊢ (¬ (𝐽0‘⟨𝐴, 𝐵⟩) ∈ Fin ↔ ω ≼ (𝐽0‘⟨𝐴, 𝐵⟩))
30	29	biimpri 231	. . . . . . . . . . 11 ⊢ (ω ≼ (𝐽0‘⟨𝐴, 𝐵⟩) → ¬ (𝐽0‘⟨𝐴, 𝐵⟩) ∈ Fin)
31	26, 30	nsyli 158	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (ω ≼ (𝐽0‘⟨𝐴, 𝐵⟩) → ¬ suc (𝐴 ∪ 𝐵) ∈ Fin))
32		onun2 6473	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ 𝐵) ∈ On)
33		onsuc 7824	. . . . . . . . . . . 12 ⊢ ((𝐴 ∪ 𝐵) ∈ On → suc (𝐴 ∪ 𝐵) ∈ On)
34		onenon 10031	. . . . . . . . . . . 12 ⊢ (suc (𝐴 ∪ 𝐵) ∈ On → suc (𝐴 ∪ 𝐵) ∈ dom card)
35	32, 33, 34	3syl 19	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → suc (𝐴 ∪ 𝐵) ∈ dom card)
36		infinfnum 35806	. . . . . . . . . . 11 ⊢ (suc (𝐴 ∪ 𝐵) ∈ dom card → (¬ suc (𝐴 ∪ 𝐵) ∈ Fin ↔ ω ≼ suc (𝐴 ∪ 𝐵)))
37	35, 36	syl 18	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (¬ suc (𝐴 ∪ 𝐵) ∈ Fin ↔ ω ≼ suc (𝐴 ∪ 𝐵)))
38	31, 37	sylibd 242	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (ω ≼ (𝐽0‘⟨𝐴, 𝐵⟩) → ω ≼ suc (𝐴 ∪ 𝐵)))
39	19, 38	syl 18	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℵ‘𝐶) ∧ 𝐵 ∈ (ℵ‘𝐶) ∧ 𝐶 ∈ On) → (ω ≼ (𝐽0‘⟨𝐴, 𝐵⟩) → ω ≼ suc (𝐴 ∪ 𝐵)))
40		infxpidm2 10097	. . . . . . . . . . . . . 14 ⊢ ((suc (𝐴 ∪ 𝐵) ∈ dom card ∧ ω ≼ suc (𝐴 ∪ 𝐵)) → (suc (𝐴 ∪ 𝐵) × suc (𝐴 ∪ 𝐵)) ≈ suc (𝐴 ∪ 𝐵))
41	35, 40	sylan 592	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ω ≼ suc (𝐴 ∪ 𝐵)) → (suc (𝐴 ∪ 𝐵) × suc (𝐴 ∪ 𝐵)) ≈ suc (𝐴 ∪ 𝐵))
42	41	ex 418	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (ω ≼ suc (𝐴 ∪ 𝐵) → (suc (𝐴 ∪ 𝐵) × suc (𝐴 ∪ 𝐵)) ≈ suc (𝐴 ∪ 𝐵)))
43		domentr 9041	. . . . . . . . . . . 12 ⊢ (((𝐽0‘⟨𝐴, 𝐵⟩) ≼ (suc (𝐴 ∪ 𝐵) × suc (𝐴 ∪ 𝐵)) ∧ (suc (𝐴 ∪ 𝐵) × suc (𝐴 ∪ 𝐵)) ≈ suc (𝐴 ∪ 𝐵)) → (𝐽0‘⟨𝐴, 𝐵⟩) ≼ suc (𝐴 ∪ 𝐵))
44	20, 42, 43	syl6an 697	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (ω ≼ suc (𝐴 ∪ 𝐵) → (𝐽0‘⟨𝐴, 𝐵⟩) ≼ suc (𝐴 ∪ 𝐵)))
45	19, 44	syl 18	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (ℵ‘𝐶) ∧ 𝐵 ∈ (ℵ‘𝐶) ∧ 𝐶 ∈ On) → (ω ≼ suc (𝐴 ∪ 𝐵) → (𝐽0‘⟨𝐴, 𝐵⟩) ≼ suc (𝐴 ∪ 𝐵)))
46		onunel 6470	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ (ℵ‘𝐶) ∈ On) → ((𝐴 ∪ 𝐵) ∈ (ℵ‘𝐶) ↔ (𝐴 ∈ (ℵ‘𝐶) ∧ 𝐵 ∈ (ℵ‘𝐶))))
47	15, 46	mp3an3 1479	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ∪ 𝐵) ∈ (ℵ‘𝐶) ↔ (𝐴 ∈ (ℵ‘𝐶) ∧ 𝐵 ∈ (ℵ‘𝐶))))
48	16, 17, 47	syl2an 608	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (ℵ‘𝐶) ∧ 𝐵 ∈ (ℵ‘𝐶)) → ((𝐴 ∪ 𝐵) ∈ (ℵ‘𝐶) ↔ (𝐴 ∈ (ℵ‘𝐶) ∧ 𝐵 ∈ (ℵ‘𝐶))))
49	48	ibir 271	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (ℵ‘𝐶) ∧ 𝐵 ∈ (ℵ‘𝐶)) → (𝐴 ∪ 𝐵) ∈ (ℵ‘𝐶))
50	49	3adant3 1150	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (ℵ‘𝐶) ∧ 𝐵 ∈ (ℵ‘𝐶) ∧ 𝐶 ∈ On) → (𝐴 ∪ 𝐵) ∈ (ℵ‘𝐶))
51		alephislim 10163	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ On ↔ Lim (ℵ‘𝐶))
52		limsuc 7860	. . . . . . . . . . . . . 14 ⊢ (Lim (ℵ‘𝐶) → ((𝐴 ∪ 𝐵) ∈ (ℵ‘𝐶) ↔ suc (𝐴 ∪ 𝐵) ∈ (ℵ‘𝐶)))
53	51, 52	sylbi 220	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ On → ((𝐴 ∪ 𝐵) ∈ (ℵ‘𝐶) ↔ suc (𝐴 ∪ 𝐵) ∈ (ℵ‘𝐶)))
54	53	3ad2ant3 1153	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (ℵ‘𝐶) ∧ 𝐵 ∈ (ℵ‘𝐶) ∧ 𝐶 ∈ On) → ((𝐴 ∪ 𝐵) ∈ (ℵ‘𝐶) ↔ suc (𝐴 ∪ 𝐵) ∈ (ℵ‘𝐶)))
55	50, 54	mpbid 235	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (ℵ‘𝐶) ∧ 𝐵 ∈ (ℵ‘𝐶) ∧ 𝐶 ∈ On) → suc (𝐴 ∪ 𝐵) ∈ (ℵ‘𝐶))
56	18, 32, 33	3syl 19	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (ℵ‘𝐶) ∧ 𝐵 ∈ (ℵ‘𝐶)) → suc (𝐴 ∪ 𝐵) ∈ On)
57		alephsdom 10166	. . . . . . . . . . . 12 ⊢ ((suc (𝐴 ∪ 𝐵) ∈ On ∧ 𝐶 ∈ On) → (suc (𝐴 ∪ 𝐵) ∈ (ℵ‘𝐶) ↔ suc (𝐴 ∪ 𝐵) ≺ (ℵ‘𝐶)))
58	56, 57	stoic3 1808	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (ℵ‘𝐶) ∧ 𝐵 ∈ (ℵ‘𝐶) ∧ 𝐶 ∈ On) → (suc (𝐴 ∪ 𝐵) ∈ (ℵ‘𝐶) ↔ suc (𝐴 ∪ 𝐵) ≺ (ℵ‘𝐶)))
59	55, 58	mpbid 235	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (ℵ‘𝐶) ∧ 𝐵 ∈ (ℵ‘𝐶) ∧ 𝐶 ∈ On) → suc (𝐴 ∪ 𝐵) ≺ (ℵ‘𝐶))
60	45, 59	jctird 536	. . . . . . . . 9 ⊢ ((𝐴 ∈ (ℵ‘𝐶) ∧ 𝐵 ∈ (ℵ‘𝐶) ∧ 𝐶 ∈ On) → (ω ≼ suc (𝐴 ∪ 𝐵) → ((𝐽0‘⟨𝐴, 𝐵⟩) ≼ suc (𝐴 ∪ 𝐵) ∧ suc (𝐴 ∪ 𝐵) ≺ (ℵ‘𝐶))))
61		domsdomtr 9132	. . . . . . . . 9 ⊢ (((𝐽0‘⟨𝐴, 𝐵⟩) ≼ suc (𝐴 ∪ 𝐵) ∧ suc (𝐴 ∪ 𝐵) ≺ (ℵ‘𝐶)) → (𝐽0‘⟨𝐴, 𝐵⟩) ≺ (ℵ‘𝐶))
62	60, 61	syl6 36	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℵ‘𝐶) ∧ 𝐵 ∈ (ℵ‘𝐶) ∧ 𝐶 ∈ On) → (ω ≼ suc (𝐴 ∪ 𝐵) → (𝐽0‘⟨𝐴, 𝐵⟩) ≺ (ℵ‘𝐶)))
63	39, 62	syld 48	. . . . . . 7 ⊢ ((𝐴 ∈ (ℵ‘𝐶) ∧ 𝐵 ∈ (ℵ‘𝐶) ∧ 𝐶 ∈ On) → (ω ≼ (𝐽0‘⟨𝐴, 𝐵⟩) → (𝐽0‘⟨𝐴, 𝐵⟩) ≺ (ℵ‘𝐶)))
64		alephsdom 10166	. . . . . . . . 9 ⊢ (((𝐽0‘⟨𝐴, 𝐵⟩) ∈ On ∧ 𝐶 ∈ On) → ((𝐽0‘⟨𝐴, 𝐵⟩) ∈ (ℵ‘𝐶) ↔ (𝐽0‘⟨𝐴, 𝐵⟩) ≺ (ℵ‘𝐶)))
65	7, 64	mpan 703	. . . . . . . 8 ⊢ (𝐶 ∈ On → ((𝐽0‘⟨𝐴, 𝐵⟩) ∈ (ℵ‘𝐶) ↔ (𝐽0‘⟨𝐴, 𝐵⟩) ≺ (ℵ‘𝐶)))
66	65	3ad2ant3 1153	. . . . . . 7 ⊢ ((𝐴 ∈ (ℵ‘𝐶) ∧ 𝐵 ∈ (ℵ‘𝐶) ∧ 𝐶 ∈ On) → ((𝐽0‘⟨𝐴, 𝐵⟩) ∈ (ℵ‘𝐶) ↔ (𝐽0‘⟨𝐴, 𝐵⟩) ≺ (ℵ‘𝐶)))
67	63, 66	sylibrd 262	. . . . . 6 ⊢ ((𝐴 ∈ (ℵ‘𝐶) ∧ 𝐵 ∈ (ℵ‘𝐶) ∧ 𝐶 ∈ On) → (ω ≼ (𝐽0‘⟨𝐴, 𝐵⟩) → (𝐽0‘⟨𝐴, 𝐵⟩) ∈ (ℵ‘𝐶)))
68		alephgeom 10162	. . . . . . . . . 10 ⊢ (𝐶 ∈ On ↔ ω ⊆ (ℵ‘𝐶))
69	68	biimpi 219	. . . . . . . . 9 ⊢ (𝐶 ∈ On → ω ⊆ (ℵ‘𝐶))
70	69	sselda 3908	. . . . . . . 8 ⊢ ((𝐶 ∈ On ∧ (𝐽0‘⟨𝐴, 𝐵⟩) ∈ ω) → (𝐽0‘⟨𝐴, 𝐵⟩) ∈ (ℵ‘𝐶))
71	70	3ad2antl3 1206	. . . . . . 7 ⊢ (((𝐴 ∈ (ℵ‘𝐶) ∧ 𝐵 ∈ (ℵ‘𝐶) ∧ 𝐶 ∈ On) ∧ (𝐽0‘⟨𝐴, 𝐵⟩) ∈ ω) → (𝐽0‘⟨𝐴, 𝐵⟩) ∈ (ℵ‘𝐶))
72	71	ex 418	. . . . . 6 ⊢ ((𝐴 ∈ (ℵ‘𝐶) ∧ 𝐵 ∈ (ℵ‘𝐶) ∧ 𝐶 ∈ On) → ((𝐽0‘⟨𝐴, 𝐵⟩) ∈ ω → (𝐽0‘⟨𝐴, 𝐵⟩) ∈ (ℵ‘𝐶)))
73	67, 72	jaod 873	. . . . 5 ⊢ ((𝐴 ∈ (ℵ‘𝐶) ∧ 𝐵 ∈ (ℵ‘𝐶) ∧ 𝐶 ∈ On) → ((ω ≼ (𝐽0‘⟨𝐴, 𝐵⟩) ∨ (𝐽0‘⟨𝐴, 𝐵⟩) ∈ ω) → (𝐽0‘⟨𝐴, 𝐵⟩) ∈ (ℵ‘𝐶)))
74	14, 73	mpi 21	. . . 4 ⊢ ((𝐴 ∈ (ℵ‘𝐶) ∧ 𝐵 ∈ (ℵ‘𝐶) ∧ 𝐶 ∈ On) → (𝐽0‘⟨𝐴, 𝐵⟩) ∈ (ℵ‘𝐶))
75	74	3expia 1139	. . 3 ⊢ ((𝐴 ∈ (ℵ‘𝐶) ∧ 𝐵 ∈ (ℵ‘𝐶)) → (𝐶 ∈ On → (𝐽0‘⟨𝐴, 𝐵⟩) ∈ (ℵ‘𝐶)))
76	75	com12 33	. 2 ⊢ (𝐶 ∈ On → ((𝐴 ∈ (ℵ‘𝐶) ∧ 𝐵 ∈ (ℵ‘𝐶)) → (𝐽0‘⟨𝐴, 𝐵⟩) ∈ (ℵ‘𝐶)))
77		alephfnon 10145	. . . . . . . 8 ⊢ ℵ Fn On
78	77	fndmi 6643	. . . . . . 7 ⊢ dom ℵ = On
79	78	eleq2i 2852	. . . . . 6 ⊢ (𝐶 ∈ dom ℵ ↔ 𝐶 ∈ On)
80		ndmfv 6917	. . . . . 6 ⊢ (¬ 𝐶 ∈ dom ℵ → (ℵ‘𝐶) = ∅)
81	79, 80	sylnbir 334	. . . . 5 ⊢ (¬ 𝐶 ∈ On → (ℵ‘𝐶) = ∅)
82		nel02 4284	. . . . 5 ⊢ ((ℵ‘𝐶) = ∅ → ¬ 𝐴 ∈ (ℵ‘𝐶))
83	81, 82	syl 18	. . . 4 ⊢ (¬ 𝐶 ∈ On → ¬ 𝐴 ∈ (ℵ‘𝐶))
84	83	intnanrd 495	. . 3 ⊢ (¬ 𝐶 ∈ On → ¬ (𝐴 ∈ (ℵ‘𝐶) ∧ 𝐵 ∈ (ℵ‘𝐶)))
85	84	pm2.21d 122	. 2 ⊢ (¬ 𝐶 ∈ On → ((𝐴 ∈ (ℵ‘𝐶) ∧ 𝐵 ∈ (ℵ‘𝐶)) → (𝐽0‘⟨𝐴, 𝐵⟩) ∈ (ℵ‘𝐶)))
86	76, 85	pm2.61i 184	1 ⊢ ((𝐴 ∈ (ℵ‘𝐶) ∧ 𝐵 ∈ (ℵ‘𝐶)) → (𝐽0‘⟨𝐴, 𝐵⟩) ∈ (ℵ‘𝐶))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ⊆ wss 3895   ∪ cun 4149  ∅c0 4278  ⟨cop 4589   class class class wbr 5102   E cep 5549   × cxp 5648  dom cdm 5650  Oncon0 6362  Lim wlim 6363  suc csuc 6364  ⟶wf 6534  –1-1-onto→wf1o 6537  ‘cfv 6538   Isom wiso 6539  ωcom 7877   ≈ cen 8971   ≼ cdom 8972   ≺ csdm 8973  Fincfn 8974  cardccrd 10017  ℵcale 10018  𝑅₀cr0 35913  𝐽₀ccj0 35914

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 5326  ax-pr 5390  ax-un 7751  ax-inf2 9643

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 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  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 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-rpss 7739  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8300  df-wrecs 8331  df-recs 8380  df-rdg 8419  df-seqom 8459  df-1o 8477  df-2o 8478  df-er 8718  df-map 8850  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-oi 9505  df-har 9552  df-wdom 9560  df-dju 9983  df-card 10021  df-aleph 10022  df-fin2 10365  df-fin4 10366  df-fin3 10367  df-fin5 10368  df-fin6 10369  df-fin7 10370  df-lexo 35921  df-r0 35922  df-j0 35923

This theorem is used by: (None)