Theorem naddssim 8621

Description: Ordinal less-than-or-equal is preserved by natural addition. (Contributed by Scott Fenton, 7-Sep-2024.)

Assertion

Ref	Expression
naddssim	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 +no 𝐶) ⊆ (𝐵 +no 𝐶)))

Proof of Theorem naddssim

Dummy variables 𝑐 𝑑 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7375	. . . . . . 7 ⊢ (𝑐 = 𝑑 → (𝐴 +no 𝑐) = (𝐴 +no 𝑑))
2		oveq2 7375	. . . . . . 7 ⊢ (𝑐 = 𝑑 → (𝐵 +no 𝑐) = (𝐵 +no 𝑑))
3	1, 2	sseq12d 3955	. . . . . 6 ⊢ (𝑐 = 𝑑 → ((𝐴 +no 𝑐) ⊆ (𝐵 +no 𝑐) ↔ (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)))
4	3	imbi2d 340	. . . . 5 ⊢ (𝑐 = 𝑑 → ((𝐴 ⊆ 𝐵 → (𝐴 +no 𝑐) ⊆ (𝐵 +no 𝑐)) ↔ (𝐴 ⊆ 𝐵 → (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑))))
5	4	imbi2d 340	. . . 4 ⊢ (𝑐 = 𝑑 → (((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 +no 𝑐) ⊆ (𝐵 +no 𝑐))) ↔ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)))))
6		oveq2 7375	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (𝐴 +no 𝑐) = (𝐴 +no 𝐶))
7		oveq2 7375	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (𝐵 +no 𝑐) = (𝐵 +no 𝐶))
8	6, 7	sseq12d 3955	. . . . . 6 ⊢ (𝑐 = 𝐶 → ((𝐴 +no 𝑐) ⊆ (𝐵 +no 𝑐) ↔ (𝐴 +no 𝐶) ⊆ (𝐵 +no 𝐶)))
9	8	imbi2d 340	. . . . 5 ⊢ (𝑐 = 𝐶 → ((𝐴 ⊆ 𝐵 → (𝐴 +no 𝑐) ⊆ (𝐵 +no 𝑐)) ↔ (𝐴 ⊆ 𝐵 → (𝐴 +no 𝐶) ⊆ (𝐵 +no 𝐶))))
10	9	imbi2d 340	. . . 4 ⊢ (𝑐 = 𝐶 → (((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 +no 𝑐) ⊆ (𝐵 +no 𝑐))) ↔ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 +no 𝐶) ⊆ (𝐵 +no 𝐶)))))
11		r19.21v 3162	. . . . . 6 ⊢ (∀𝑑 ∈ 𝑐 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑))) ↔ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∀𝑑 ∈ 𝑐 (𝐴 ⊆ 𝐵 → (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑))))
12		r19.21v 3162	. . . . . . 7 ⊢ (∀𝑑 ∈ 𝑐 (𝐴 ⊆ 𝐵 → (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) ↔ (𝐴 ⊆ 𝐵 → ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)))
13	12	imbi2i 336	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∀𝑑 ∈ 𝑐 (𝐴 ⊆ 𝐵 → (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑))) ↔ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑))))
14	11, 13	bitri 275	. . . . 5 ⊢ (∀𝑑 ∈ 𝑐 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑))) ↔ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑))))
15		oveq2 7375	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑑 = 𝑤 → (𝐴 +no 𝑑) = (𝐴 +no 𝑤))
16		oveq2 7375	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑑 = 𝑤 → (𝐵 +no 𝑑) = (𝐵 +no 𝑤))
17	15, 16	sseq12d 3955	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = 𝑤 → ((𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑) ↔ (𝐴 +no 𝑤) ⊆ (𝐵 +no 𝑤)))
18	17	rspccva 3563	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑) ∧ 𝑤 ∈ 𝑐) → (𝐴 +no 𝑤) ⊆ (𝐵 +no 𝑤))
19	18	ad4ant24 755	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) ∧ (𝑥 ∈ On ∧ (∀𝑦 ∈ 𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (𝑧 +no 𝑐) ∈ 𝑥))) ∧ 𝑤 ∈ 𝑐) → (𝐴 +no 𝑤) ⊆ (𝐵 +no 𝑤))
20		simprrl 781	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) ∧ (𝑥 ∈ On ∧ (∀𝑦 ∈ 𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (𝑧 +no 𝑐) ∈ 𝑥))) → ∀𝑦 ∈ 𝑐 (𝐵 +no 𝑦) ∈ 𝑥)
21		oveq2 7375	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑤 → (𝐵 +no 𝑦) = (𝐵 +no 𝑤))
22	21	eleq1d 2821	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑤 → ((𝐵 +no 𝑦) ∈ 𝑥 ↔ (𝐵 +no 𝑤) ∈ 𝑥))
23	22	rspccva 3563	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑦 ∈ 𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ 𝑤 ∈ 𝑐) → (𝐵 +no 𝑤) ∈ 𝑥)
24	20, 23	sylan 581	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) ∧ (𝑥 ∈ On ∧ (∀𝑦 ∈ 𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (𝑧 +no 𝑐) ∈ 𝑥))) ∧ 𝑤 ∈ 𝑐) → (𝐵 +no 𝑤) ∈ 𝑥)
25		simplrl 777	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ On)
26	25	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) → 𝐴 ∈ On)
27	26	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) ∧ (𝑥 ∈ On ∧ (∀𝑦 ∈ 𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (𝑧 +no 𝑐) ∈ 𝑥))) → 𝐴 ∈ On)
28	27	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) ∧ (𝑥 ∈ On ∧ (∀𝑦 ∈ 𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (𝑧 +no 𝑐) ∈ 𝑥))) ∧ 𝑤 ∈ 𝑐) → 𝐴 ∈ On)
29		simp-4l 783	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) ∧ (𝑥 ∈ On ∧ (∀𝑦 ∈ 𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (𝑧 +no 𝑐) ∈ 𝑥))) → 𝑐 ∈ On)
30		onelon 6348	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑐 ∈ On ∧ 𝑤 ∈ 𝑐) → 𝑤 ∈ On)
31	29, 30	sylan 581	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) ∧ (𝑥 ∈ On ∧ (∀𝑦 ∈ 𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (𝑧 +no 𝑐) ∈ 𝑥))) ∧ 𝑤 ∈ 𝑐) → 𝑤 ∈ On)
32		naddcl 8613	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ On ∧ 𝑤 ∈ On) → (𝐴 +no 𝑤) ∈ On)
33	28, 31, 32	syl2anc 585	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) ∧ (𝑥 ∈ On ∧ (∀𝑦 ∈ 𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (𝑧 +no 𝑐) ∈ 𝑥))) ∧ 𝑤 ∈ 𝑐) → (𝐴 +no 𝑤) ∈ On)
34		simplrl 777	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) ∧ (𝑥 ∈ On ∧ (∀𝑦 ∈ 𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (𝑧 +no 𝑐) ∈ 𝑥))) ∧ 𝑤 ∈ 𝑐) → 𝑥 ∈ On)
35		ontr2 6371	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 +no 𝑤) ∈ On ∧ 𝑥 ∈ On) → (((𝐴 +no 𝑤) ⊆ (𝐵 +no 𝑤) ∧ (𝐵 +no 𝑤) ∈ 𝑥) → (𝐴 +no 𝑤) ∈ 𝑥))
36	33, 34, 35	syl2anc 585	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) ∧ (𝑥 ∈ On ∧ (∀𝑦 ∈ 𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (𝑧 +no 𝑐) ∈ 𝑥))) ∧ 𝑤 ∈ 𝑐) → (((𝐴 +no 𝑤) ⊆ (𝐵 +no 𝑤) ∧ (𝐵 +no 𝑤) ∈ 𝑥) → (𝐴 +no 𝑤) ∈ 𝑥))
37	19, 24, 36	mp2and 700	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) ∧ (𝑥 ∈ On ∧ (∀𝑦 ∈ 𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (𝑧 +no 𝑐) ∈ 𝑥))) ∧ 𝑤 ∈ 𝑐) → (𝐴 +no 𝑤) ∈ 𝑥)
38	37	ralrimiva 3129	. . . . . . . . . . . . . 14 ⊢ (((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) ∧ (𝑥 ∈ On ∧ (∀𝑦 ∈ 𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (𝑧 +no 𝑐) ∈ 𝑥))) → ∀𝑤 ∈ 𝑐 (𝐴 +no 𝑤) ∈ 𝑥)
39		simpllr 776	. . . . . . . . . . . . . . 15 ⊢ (((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) ∧ (𝑥 ∈ On ∧ (∀𝑦 ∈ 𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (𝑧 +no 𝑐) ∈ 𝑥))) → 𝐴 ⊆ 𝐵)
40		simprrr 782	. . . . . . . . . . . . . . 15 ⊢ (((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) ∧ (𝑥 ∈ On ∧ (∀𝑦 ∈ 𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (𝑧 +no 𝑐) ∈ 𝑥))) → ∀𝑧 ∈ 𝐵 (𝑧 +no 𝑐) ∈ 𝑥)
41		ssralv 3990	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑧 ∈ 𝐵 (𝑧 +no 𝑐) ∈ 𝑥 → ∀𝑧 ∈ 𝐴 (𝑧 +no 𝑐) ∈ 𝑥))
42	39, 40, 41	sylc 65	. . . . . . . . . . . . . 14 ⊢ (((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) ∧ (𝑥 ∈ On ∧ (∀𝑦 ∈ 𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (𝑧 +no 𝑐) ∈ 𝑥))) → ∀𝑧 ∈ 𝐴 (𝑧 +no 𝑐) ∈ 𝑥)
43	38, 42	jca 511	. . . . . . . . . . . . 13 ⊢ (((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) ∧ (𝑥 ∈ On ∧ (∀𝑦 ∈ 𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (𝑧 +no 𝑐) ∈ 𝑥))) → (∀𝑤 ∈ 𝑐 (𝐴 +no 𝑤) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝐴 (𝑧 +no 𝑐) ∈ 𝑥))
44	43	expr 456	. . . . . . . . . . . 12 ⊢ (((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) ∧ 𝑥 ∈ On) → ((∀𝑦 ∈ 𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (𝑧 +no 𝑐) ∈ 𝑥) → (∀𝑤 ∈ 𝑐 (𝐴 +no 𝑤) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝐴 (𝑧 +no 𝑐) ∈ 𝑥)))
45	44	ss2rabdv 4015	. . . . . . . . . . 11 ⊢ ((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) → {𝑥 ∈ On ∣ (∀𝑦 ∈ 𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (𝑧 +no 𝑐) ∈ 𝑥)} ⊆ {𝑥 ∈ On ∣ (∀𝑤 ∈ 𝑐 (𝐴 +no 𝑤) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝐴 (𝑧 +no 𝑐) ∈ 𝑥)})
46		intss 4911	. . . . . . . . . . 11 ⊢ ({𝑥 ∈ On ∣ (∀𝑦 ∈ 𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (𝑧 +no 𝑐) ∈ 𝑥)} ⊆ {𝑥 ∈ On ∣ (∀𝑤 ∈ 𝑐 (𝐴 +no 𝑤) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝐴 (𝑧 +no 𝑐) ∈ 𝑥)} → ∩ {𝑥 ∈ On ∣ (∀𝑤 ∈ 𝑐 (𝐴 +no 𝑤) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝐴 (𝑧 +no 𝑐) ∈ 𝑥)} ⊆ ∩ {𝑥 ∈ On ∣ (∀𝑦 ∈ 𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (𝑧 +no 𝑐) ∈ 𝑥)})
47	45, 46	syl 17	. . . . . . . . . 10 ⊢ ((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) → ∩ {𝑥 ∈ On ∣ (∀𝑤 ∈ 𝑐 (𝐴 +no 𝑤) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝐴 (𝑧 +no 𝑐) ∈ 𝑥)} ⊆ ∩ {𝑥 ∈ On ∣ (∀𝑦 ∈ 𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (𝑧 +no 𝑐) ∈ 𝑥)})
48		simplll 775	. . . . . . . . . . 11 ⊢ ((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) → 𝑐 ∈ On)
49		naddov2 8615	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑐 ∈ On) → (𝐴 +no 𝑐) = ∩ {𝑥 ∈ On ∣ (∀𝑤 ∈ 𝑐 (𝐴 +no 𝑤) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝐴 (𝑧 +no 𝑐) ∈ 𝑥)})
50	26, 48, 49	syl2anc 585	. . . . . . . . . 10 ⊢ ((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) → (𝐴 +no 𝑐) = ∩ {𝑥 ∈ On ∣ (∀𝑤 ∈ 𝑐 (𝐴 +no 𝑤) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝐴 (𝑧 +no 𝑐) ∈ 𝑥)})
51		simplrr 778	. . . . . . . . . . . 12 ⊢ (((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) → 𝐵 ∈ On)
52	51	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) → 𝐵 ∈ On)
53		naddov2 8615	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝑐 ∈ On) → (𝐵 +no 𝑐) = ∩ {𝑥 ∈ On ∣ (∀𝑦 ∈ 𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (𝑧 +no 𝑐) ∈ 𝑥)})
54	52, 48, 53	syl2anc 585	. . . . . . . . . 10 ⊢ ((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) → (𝐵 +no 𝑐) = ∩ {𝑥 ∈ On ∣ (∀𝑦 ∈ 𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (𝑧 +no 𝑐) ∈ 𝑥)})
55	47, 50, 54	3sstr4d 3977	. . . . . . . . 9 ⊢ ((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) → (𝐴 +no 𝑐) ⊆ (𝐵 +no 𝑐))
56	55	exp31 419	. . . . . . . 8 ⊢ ((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ⊆ 𝐵 → (∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑) → (𝐴 +no 𝑐) ⊆ (𝐵 +no 𝑐))))
57	56	a2d 29	. . . . . . 7 ⊢ ((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → ((𝐴 ⊆ 𝐵 → ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) → (𝐴 ⊆ 𝐵 → (𝐴 +no 𝑐) ⊆ (𝐵 +no 𝑐))))
58	57	ex 412	. . . . . 6 ⊢ (𝑐 ∈ On → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ⊆ 𝐵 → ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) → (𝐴 ⊆ 𝐵 → (𝐴 +no 𝑐) ⊆ (𝐵 +no 𝑐)))))
59	58	a2d 29	. . . . 5 ⊢ (𝑐 ∈ On → (((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑))) → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 +no 𝑐) ⊆ (𝐵 +no 𝑐)))))
60	14, 59	biimtrid 242	. . . 4 ⊢ (𝑐 ∈ On → (∀𝑑 ∈ 𝑐 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑))) → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 +no 𝑐) ⊆ (𝐵 +no 𝑐)))))
61	5, 10, 60	tfis3 7809	. . 3 ⊢ (𝐶 ∈ On → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 +no 𝐶) ⊆ (𝐵 +no 𝐶))))
62	61	com12 32	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ∈ On → (𝐴 ⊆ 𝐵 → (𝐴 +no 𝐶) ⊆ (𝐵 +no 𝐶))))
63	62	3impia 1118	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 +no 𝐶) ⊆ (𝐵 +no 𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3051  {crab 3389   ⊆ wss 3889  ∩ cint 4889  Oncon0 6323  (class class class)co 7367   +no cnadd 8601

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-frecs 8231  df-nadd 8602

This theorem is referenced by:  naddel1  8623  nadd2rabex  43814