Theorem j0ltalephlem 36007

Description: Lemma for j0ltaleph 36008. (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 6921	. . 3 ⊢ (𝐽0‘⟨𝐴, 𝐵⟩) ∈ V
2		j0iso 36004	. . . . 5 ⊢ 𝐽0 Isom 𝑅0, E ((On × On), On)
3		opelxpi 5692	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ⟨𝐴, 𝐵⟩ ∈ (On × On))
4		imassrn 6210	. . . . . . . . 9 ⊢ (◡𝑅0 “ {⟨𝐴, 𝐵⟩}) ⊆ ran ◡𝑅0
5		dfdm4 5884	. . . . . . . . . 10 ⊢ dom 𝑅0 = ran ◡𝑅0
6		dmr0ss 36002	. . . . . . . . . 10 ⊢ dom 𝑅0 ⊆ (On × On)
7	5, 6	eqsstrri 3960	. . . . . . . . 9 ⊢ ran ◡𝑅0 ⊆ (On × On)
8	4, 7	sstri 3922	. . . . . . . 8 ⊢ (◡𝑅0 “ {⟨𝐴, 𝐵⟩}) ⊆ (On × On)
9		sseqin2 4123	. . . . . . . 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 2772	. . . . . 6 ⊢ (◡𝑅0 “ {⟨𝐴, 𝐵⟩}) = ((On × On) ∩ (◡𝑅0 “ {⟨𝐴, 𝐵⟩}))
12	1	epini 6106	. . . . . . . 8 ⊢ (◡ E “ {(𝐽0‘⟨𝐴, 𝐵⟩)}) = (𝐽0‘⟨𝐴, 𝐵⟩)
13	12	ineq2i 4117	. . . . . . 7 ⊢ (On ∩ (◡ E “ {(𝐽0‘⟨𝐴, 𝐵⟩)})) = (On ∩ (𝐽0‘⟨𝐴, 𝐵⟩))
14		isof1o 7354	. . . . . . . . . . 11 ⊢ (𝐽0 Isom 𝑅0, E ((On × On), On) → 𝐽0:(On × On)⟶1-1ontoOn)
15		f1of 6846	. . . . . . . . . . 11 ⊢ (𝐽0:(On × On)⟶1-1ontoOn → 𝐽0:(On × On)⟶On)
16	2, 14, 15	mp2b 10	. . . . . . . . . 10 ⊢ 𝐽0:(On × On)⟶On
17		0elon 6436	. . . . . . . . . 10 ⊢ ∅ ∈ On
18	16, 17	f0cli 7122	. . . . . . . . 9 ⊢ (𝐽0‘⟨𝐴, 𝐵⟩) ∈ On
19	18	onssi 7873	. . . . . . . 8 ⊢ (𝐽0‘⟨𝐴, 𝐵⟩) ⊆ On
20		sseqin2 4123	. . . . . . . 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 2787	. . . . . 6 ⊢ (𝐽0‘⟨𝐴, 𝐵⟩) = (On ∩ (◡ E “ {(𝐽0‘⟨𝐴, 𝐵⟩)}))
23	11, 22	isoini2 7370	. . . . 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 7354	. . . 4 ⊢ ((𝐽0 ↾ (◡𝑅0 “ {⟨𝐴, 𝐵⟩})) Isom 𝑅0, E ((◡𝑅0 “ {⟨𝐴, 𝐵⟩}), (𝐽0‘⟨𝐴, 𝐵⟩)) → (𝐽0 ↾ (◡𝑅0 “ {⟨𝐴, 𝐵⟩})):(◡𝑅0 “ {⟨𝐴, 𝐵⟩})⟶1-1onto(𝐽0‘⟨𝐴, 𝐵⟩))
26		f1ocnv 6859	. . . 4 ⊢ ((𝐽0 ↾ (◡𝑅0 “ {⟨𝐴, 𝐵⟩})):(◡𝑅0 “ {⟨𝐴, 𝐵⟩})⟶1-1onto(𝐽0‘⟨𝐴, 𝐵⟩) → ◡(𝐽0 ↾ (◡𝑅0 “ {⟨𝐴, 𝐵⟩})):(𝐽0‘⟨𝐴, 𝐵⟩)⟶1-1onto(◡𝑅0 “ {⟨𝐴, 𝐵⟩}))
27	24, 25, 26	3syl 19	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ◡(𝐽0 ↾ (◡𝑅0 “ {⟨𝐴, 𝐵⟩})):(𝐽0‘⟨𝐴, 𝐵⟩)⟶1-1onto(◡𝑅0 “ {⟨𝐴, 𝐵⟩}))
28		f1oeng 9025	. . 3 ⊢ (((𝐽0‘⟨𝐴, 𝐵⟩) ∈ V ∧ ◡(𝐽0 ↾ (◡𝑅0 “ {⟨𝐴, 𝐵⟩})):(𝐽0‘⟨𝐴, 𝐵⟩)⟶1-1onto(◡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 6491	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ 𝐵) ∈ On)
31		onsuc 7848	. . . . 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 7789	. . 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 5436	. . . . . . . . . . . 12 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
39		opex 5436	. . . . . . . . . . . . 13 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
40	39	eliniseg 6104	. . . . . . . . . . . 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 36003	. . . . . . . . 9 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ On)) ∧ ⟨𝑥, 𝑦⟩ ∈ (◡𝑅0 “ {⟨𝐴, 𝐵⟩})) → (𝑥 ∪ 𝑦) ⊆ (𝐴 ∪ 𝐵))
44	43	unssad 4194	. . . . . . . 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 6473	. . . . . . . . 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 4195	. . . . . . . 8 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ On ∧ 𝑦 ∈ On)) ∧ ⟨𝑥, 𝑦⟩ ∈ (◡𝑅0 “ {⟨𝐴, 𝐵⟩})) → 𝑦 ⊆ (𝐴 ∪ 𝐵))
50		onsssuc 6473	. . . . . . . . 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 5694	. . . . . 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 3209	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∀𝑥 ∈ On ∀𝑦 ∈ On (⟨𝑥, 𝑦⟩ ∈ (◡𝑅0 “ {⟨𝐴, 𝐵⟩}) → ⟨𝑥, 𝑦⟩ ∈ (suc (𝐴 ∪ 𝐵) × suc (𝐴 ∪ 𝐵))))
56		ssrel2 5766	. . . . 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 9055	. . 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 9067	. 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 1570   ∈ wcel 2147  ∀wral 3080  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092   ∪ cun 4152  {csn 4588  ⟨cop 4838   class class class wbr 5106   E cep 5554   × cxp 5653  ^◡ccnv 5654  dom cdm 5655  ran crn 5656   ↾ cres 5657   “ cima 5658  Oncon0 6380  suc csuc 6382  ⟶wf 6554  ⟶1-1ontowf1o 6557  ‘cfv 6558   Isom wiso 6559   ≈ cen 8998   ≼ cdom 8999  𝑅₀cr0 35981  𝐽₀ccj0 35982

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 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

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 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-isom 6567  df-riota 7400  df-ov 7446  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-en 9002  df-dom 9003  df-oi 9531  df-lexo 35989  df-r0 35990  df-j0 35991

This theorem is used by:  j0ltaleph  36008