Theorem j0ltalephlem 35939

Description: Lemma for j0ltaleph 35940. (Contributed by BTernaryTau, 25-Sep-2026.)

Assertion

Ref	Expression
j0ltalephlem	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐽0‘⟨𝐴, 𝐵⟩) ≼ (suc (𝐴 ∪ 𝐵) × suc (𝐴 ∪ 𝐵)))

Proof of Theorem j0ltalephlem

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

Step	Hyp	Ref	Expression
1		fvex 6898	. . 3 ⊢ (𝐽0‘⟨𝐴, 𝐵⟩) ∈ V
2		j0iso 35936	. . . . 5 ⊢ 𝐽0 Isom 𝑅0, E ((On × On), On)
3		opelxpi 5687	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ⟨𝐴, 𝐵⟩ ∈ (On × On))
4		imassrn 6197	. . . . . . . . 9 ⊢ (◡𝑅0 “ {⟨𝐴, 𝐵⟩}) ⊆ ran ◡𝑅0
5		dfdm4 5876	. . . . . . . . . 10 ⊢ dom 𝑅0 = ran ◡𝑅0
6		dmr0ss 35934	. . . . . . . . . 10 ⊢ dom 𝑅0 ⊆ (On × On)
7	5, 6	eqsstrri 3955	. . . . . . . . 9 ⊢ ran ◡𝑅0 ⊆ (On × On)
8	4, 7	sstri 3917	. . . . . . . 8 ⊢ (◡𝑅0 “ {⟨𝐴, 𝐵⟩}) ⊆ (On × On)
9		sseqin2 4120	. . . . . . . 8 ⊢ ((◡𝑅0 “ {⟨𝐴, 𝐵⟩}) ⊆ (On × On) ↔ ((On × On) ∩ (◡𝑅0 “ {⟨𝐴, 𝐵⟩})) = (◡𝑅0 “ {⟨𝐴, 𝐵⟩}))
10	8, 9	mpbi 233	. . . . . . 7 ⊢ ((On × On) ∩ (◡𝑅0 “ {⟨𝐴, 𝐵⟩})) = (◡𝑅0 “ {⟨𝐴, 𝐵⟩})
11	10	eqcomi 2769	. . . . . 6 ⊢ (◡𝑅0 “ {⟨𝐴, 𝐵⟩}) = ((On × On) ∩ (◡𝑅0 “ {⟨𝐴, 𝐵⟩}))
12	1	epini 6093	. . . . . . . 8 ⊢ (◡ E “ {(𝐽0‘⟨𝐴, 𝐵⟩)}) = (𝐽0‘⟨𝐴, 𝐵⟩)
13	12	ineq2i 4114	. . . . . . 7 ⊢ (On ∩ (◡ E “ {(𝐽0‘⟨𝐴, 𝐵⟩)})) = (On ∩ (𝐽0‘⟨𝐴, 𝐵⟩))
14		isof1o 7331	. . . . . . . . . . 11 ⊢ (𝐽0 Isom 𝑅0, E ((On × On), On) → 𝐽0:(On × On)–1-1-onto→On)
15		f1of 6824	. . . . . . . . . . 11 ⊢ (𝐽0:(On × On)–1-1-onto→On → 𝐽0:(On × On)⟶On)
16	2, 14, 15	mp2b 10	. . . . . . . . . 10 ⊢ 𝐽0:(On × On)⟶On
17		0elon 6418	. . . . . . . . . 10 ⊢ ∅ ∈ On
18	16, 17	f0cli 7098	. . . . . . . . 9 ⊢ (𝐽0‘⟨𝐴, 𝐵⟩) ∈ On
19	18	onssi 7849	. . . . . . . 8 ⊢ (𝐽0‘⟨𝐴, 𝐵⟩) ⊆ On
20		sseqin2 4120	. . . . . . . 8 ⊢ ((𝐽0‘⟨𝐴, 𝐵⟩) ⊆ On ↔ (On ∩ (𝐽0‘⟨𝐴, 𝐵⟩)) = (𝐽0‘⟨𝐴, 𝐵⟩))
21	19, 20	mpbi 233	. . . . . . 7 ⊢ (On ∩ (𝐽0‘⟨𝐴, 𝐵⟩)) = (𝐽0‘⟨𝐴, 𝐵⟩)
22	13, 21	eqtr2i 2784	. . . . . 6 ⊢ (𝐽0‘⟨𝐴, 𝐵⟩) = (On ∩ (◡ E “ {(𝐽0‘⟨𝐴, 𝐵⟩)}))
23	11, 22	isoini2 7347	. . . . 5 ⊢ ((𝐽0 Isom 𝑅0, E ((On × On), On) ∧ ⟨𝐴, 𝐵⟩ ∈ (On × On)) → (𝐽0 ↾ (◡𝑅0 “ {⟨𝐴, 𝐵⟩})) Isom 𝑅0, E ((◡𝑅0 “ {⟨𝐴, 𝐵⟩}), (𝐽0‘⟨𝐴, 𝐵⟩)))
24	2, 3, 23	sylancr 599	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐽0 ↾ (◡𝑅0 “ {⟨𝐴, 𝐵⟩})) Isom 𝑅0, E ((◡𝑅0 “ {⟨𝐴, 𝐵⟩}), (𝐽0‘⟨𝐴, 𝐵⟩)))
25		isof1o 7331	. . . 4 ⊢ ((𝐽0 ↾ (◡𝑅0 “ {⟨𝐴, 𝐵⟩})) Isom 𝑅0, E ((◡𝑅0 “ {⟨𝐴, 𝐵⟩}), (𝐽0‘⟨𝐴, 𝐵⟩)) → (𝐽0 ↾ (◡𝑅0 “ {⟨𝐴, 𝐵⟩})):(◡𝑅0 “ {⟨𝐴, 𝐵⟩})–1-1-onto→(𝐽0‘⟨𝐴, 𝐵⟩))
26		f1ocnv 6837	. . . 4 ⊢ ((𝐽0 ↾ (◡𝑅0 “ {⟨𝐴, 𝐵⟩})):(◡𝑅0 “ {⟨𝐴, 𝐵⟩})–1-1-onto→(𝐽0‘⟨𝐴, 𝐵⟩) → ◡(𝐽0 ↾ (◡𝑅0 “ {⟨𝐴, 𝐵⟩})):(𝐽0‘⟨𝐴, 𝐵⟩)–1-1-onto→(◡𝑅0 “ {⟨𝐴, 𝐵⟩}))
27	24, 25, 26	3syl 19	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ◡(𝐽0 ↾ (◡𝑅0 “ {⟨𝐴, 𝐵⟩})):(𝐽0‘⟨𝐴, 𝐵⟩)–1-1-onto→(◡𝑅0 “ {⟨𝐴, 𝐵⟩}))
28		f1oeng 8998	. . 3 ⊢ (((𝐽0‘⟨𝐴, 𝐵⟩) ∈ V ∧ ◡(𝐽0 ↾ (◡𝑅0 “ {⟨𝐴, 𝐵⟩})):(𝐽0‘⟨𝐴, 𝐵⟩)–1-1-onto→(◡𝑅0 “ {⟨𝐴, 𝐵⟩})) → (𝐽0‘⟨𝐴, 𝐵⟩) ≈ (◡𝑅0 “ {⟨𝐴, 𝐵⟩}))
29	1, 27, 28	sylancr 599	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐽0‘⟨𝐴, 𝐵⟩) ≈ (◡𝑅0 “ {⟨𝐴, 𝐵⟩}))
30		onun2 6473	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ 𝐵) ∈ On)
31		onsuc 7824	. . . . 5 ⊢ ((𝐴 ∪ 𝐵) ∈ On → suc (𝐴 ∪ 𝐵) ∈ On)
32	30, 31	syl 18	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → suc (𝐴 ∪ 𝐵) ∈ On)
33	32, 32	xpexd 7765	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (suc (𝐴 ∪ 𝐵) × suc (𝐴 ∪ 𝐵)) ∈ V)
34		simplrl 789	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ On)) ∧ ⟨𝑥, 𝑦⟩ ∈ (◡𝑅0 “ {⟨𝐴, 𝐵⟩})) → 𝑥 ∈ On)
35		simplrr 790	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ On)) ∧ ⟨𝑥, 𝑦⟩ ∈ (◡𝑅0 “ {⟨𝐴, 𝐵⟩})) → 𝑦 ∈ On)
36		simplll 787	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ On)) ∧ ⟨𝑥, 𝑦⟩ ∈ (◡𝑅0 “ {⟨𝐴, 𝐵⟩})) → 𝐴 ∈ On)
37		simpllr 788	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ On)) ∧ ⟨𝑥, 𝑦⟩ ∈ (◡𝑅0 “ {⟨𝐴, 𝐵⟩})) → 𝐵 ∈ On)
38		opex 5431	. . . . . . . . . . . 12 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
39		opex 5431	. . . . . . . . . . . . 13 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
40	39	eliniseg 6091	. . . . . . . . . . . 12 ⊢ (⟨𝐴, 𝐵⟩ ∈ V → (⟨𝑥, 𝑦⟩ ∈ (◡𝑅0 “ {⟨𝐴, 𝐵⟩}) ↔ ⟨𝑥, 𝑦⟩𝑅0⟨𝐴, 𝐵⟩))
41	38, 40	ax-mp 5	. . . . . . . . . . 11 ⊢ (⟨𝑥, 𝑦⟩ ∈ (◡𝑅0 “ {⟨𝐴, 𝐵⟩}) ↔ ⟨𝑥, 𝑦⟩𝑅0⟨𝐴, 𝐵⟩)
42	41	bilani 510	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ On)) ∧ ⟨𝑥, 𝑦⟩ ∈ (◡𝑅0 “ {⟨𝐴, 𝐵⟩})) → ⟨𝑥, 𝑦⟩𝑅0⟨𝐴, 𝐵⟩)
43	34, 35, 36, 37, 42	r0maxss 35935	. . . . . . . . 9 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ On)) ∧ ⟨𝑥, 𝑦⟩ ∈ (◡𝑅0 “ {⟨𝐴, 𝐵⟩})) → (𝑥 ∪ 𝑦) ⊆ (𝐴 ∪ 𝐵))
44	43	unssad 4191	. . . . . . . 8 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ On)) ∧ ⟨𝑥, 𝑦⟩ ∈ (◡𝑅0 “ {⟨𝐴, 𝐵⟩})) → 𝑥 ⊆ (𝐴 ∪ 𝐵))
45	36, 37, 30	syl2anc 596	. . . . . . . . 9 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ On)) ∧ ⟨𝑥, 𝑦⟩ ∈ (◡𝑅0 “ {⟨𝐴, 𝐵⟩})) → (𝐴 ∪ 𝐵) ∈ On)
46		onsssuc 6455	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ (𝐴 ∪ 𝐵) ∈ On) → (𝑥 ⊆ (𝐴 ∪ 𝐵) ↔ 𝑥 ∈ suc (𝐴 ∪ 𝐵)))
47	34, 45, 46	syl2anc 596	. . . . . . . 8 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ On)) ∧ ⟨𝑥, 𝑦⟩ ∈ (◡𝑅0 “ {⟨𝐴, 𝐵⟩})) → (𝑥 ⊆ (𝐴 ∪ 𝐵) ↔ 𝑥 ∈ suc (𝐴 ∪ 𝐵)))
48	44, 47	mpbid 235	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ On)) ∧ ⟨𝑥, 𝑦⟩ ∈ (◡𝑅0 “ {⟨𝐴, 𝐵⟩})) → 𝑥 ∈ suc (𝐴 ∪ 𝐵))
49	43	unssbd 4192	. . . . . . . 8 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ On)) ∧ ⟨𝑥, 𝑦⟩ ∈ (◡𝑅0 “ {⟨𝐴, 𝐵⟩})) → 𝑦 ⊆ (𝐴 ∪ 𝐵))
50		onsssuc 6455	. . . . . . . . 9 ⊢ ((𝑦 ∈ On ∧ (𝐴 ∪ 𝐵) ∈ On) → (𝑦 ⊆ (𝐴 ∪ 𝐵) ↔ 𝑦 ∈ suc (𝐴 ∪ 𝐵)))
51	35, 45, 50	syl2anc 596	. . . . . . . 8 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ On)) ∧ ⟨𝑥, 𝑦⟩ ∈ (◡𝑅0 “ {⟨𝐴, 𝐵⟩})) → (𝑦 ⊆ (𝐴 ∪ 𝐵) ↔ 𝑦 ∈ suc (𝐴 ∪ 𝐵)))
52	49, 51	mpbid 235	. . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ On)) ∧ ⟨𝑥, 𝑦⟩ ∈ (◡𝑅0 “ {⟨𝐴, 𝐵⟩})) → 𝑦 ∈ suc (𝐴 ∪ 𝐵))
53	48, 52	opelxpd 5689	. . . . . 6 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ On)) ∧ ⟨𝑥, 𝑦⟩ ∈ (◡𝑅0 “ {⟨𝐴, 𝐵⟩})) → ⟨𝑥, 𝑦⟩ ∈ (suc (𝐴 ∪ 𝐵) × suc (𝐴 ∪ 𝐵)))
54	53	ex 418	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ On)) → (⟨𝑥, 𝑦⟩ ∈ (◡𝑅0 “ {⟨𝐴, 𝐵⟩}) → ⟨𝑥, 𝑦⟩ ∈ (suc (𝐴 ∪ 𝐵) × suc (𝐴 ∪ 𝐵))))
55	54	ralrimivva 3205	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∀𝑥 ∈ On ∀𝑦 ∈ On (⟨𝑥, 𝑦⟩ ∈ (◡𝑅0 “ {⟨𝐴, 𝐵⟩}) → ⟨𝑥, 𝑦⟩ ∈ (suc (𝐴 ∪ 𝐵) × suc (𝐴 ∪ 𝐵))))
56		ssrel2 5760	. . . . 5 ⊢ ((◡𝑅0 “ {⟨𝐴, 𝐵⟩}) ⊆ (On × On) → ((◡𝑅0 “ {⟨𝐴, 𝐵⟩}) ⊆ (suc (𝐴 ∪ 𝐵) × suc (𝐴 ∪ 𝐵)) ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On (⟨𝑥, 𝑦⟩ ∈ (◡𝑅0 “ {⟨𝐴, 𝐵⟩}) → ⟨𝑥, 𝑦⟩ ∈ (suc (𝐴 ∪ 𝐵) × suc (𝐴 ∪ 𝐵)))))
57	8, 56	ax-mp 5	. . . 4 ⊢ ((◡𝑅0 “ {⟨𝐴, 𝐵⟩}) ⊆ (suc (𝐴 ∪ 𝐵) × suc (𝐴 ∪ 𝐵)) ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On (⟨𝑥, 𝑦⟩ ∈ (◡𝑅0 “ {⟨𝐴, 𝐵⟩}) → ⟨𝑥, 𝑦⟩ ∈ (suc (𝐴 ∪ 𝐵) × suc (𝐴 ∪ 𝐵))))
58	55, 57	sylibr 237	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (◡𝑅0 “ {⟨𝐴, 𝐵⟩}) ⊆ (suc (𝐴 ∪ 𝐵) × suc (𝐴 ∪ 𝐵)))
59		ssdomg 9028	. . 3 ⊢ ((suc (𝐴 ∪ 𝐵) × suc (𝐴 ∪ 𝐵)) ∈ V → ((◡𝑅0 “ {⟨𝐴, 𝐵⟩}) ⊆ (suc (𝐴 ∪ 𝐵) × suc (𝐴 ∪ 𝐵)) → (◡𝑅0 “ {⟨𝐴, 𝐵⟩}) ≼ (suc (𝐴 ∪ 𝐵) × suc (𝐴 ∪ 𝐵))))
60	33, 58, 59	sylc 66	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (◡𝑅0 “ {⟨𝐴, 𝐵⟩}) ≼ (suc (𝐴 ∪ 𝐵) × suc (𝐴 ∪ 𝐵)))
61		endomtr 9040	. 2 ⊢ (((𝐽0‘⟨𝐴, 𝐵⟩) ≈ (◡𝑅0 “ {⟨𝐴, 𝐵⟩}) ∧ (◡𝑅0 “ {⟨𝐴, 𝐵⟩}) ≼ (suc (𝐴 ∪ 𝐵) × suc (𝐴 ∪ 𝐵))) → (𝐽0‘⟨𝐴, 𝐵⟩) ≼ (suc (𝐴 ∪ 𝐵) × suc (𝐴 ∪ 𝐵)))
62	29, 60, 61	syl2anc 596	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐽0‘⟨𝐴, 𝐵⟩) ≼ (suc (𝐴 ∪ 𝐵) × suc (𝐴 ∪ 𝐵)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  Vcvv 3450   ⊆ wss 3895   ∩ cin 4089   ∪ cun 4149  {csn 4583  ⟨cop 4589   class class class wbr 5102   E cep 5549   × cxp 5648  ^◡ccnv 5649  dom cdm 5650  ran crn 5651   ↾ cres 5652   “ cima 5653  Oncon0 6362  suc csuc 6364  ⟶wf 6534  –1-1-onto→wf1o 6537  ‘cfv 6538   Isom wiso 6539   ≈ cen 8971   ≼ cdom 8972  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

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-1st 8001  df-2nd 8002  df-frecs 8300  df-wrecs 8331  df-recs 8380  df-en 8975  df-dom 8976  df-oi 9505  df-lexo 35921  df-r0 35922  df-j0 35923

This theorem is used by:  j0ltaleph  35940