Theorem naddssim 8702

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 7432	. . . . . . 7 ⊢ (𝑐 = 𝑑 → (𝐴 +no 𝑐) = (𝐴 +no 𝑑))
2		oveq2 7432	. . . . . . 7 ⊢ (𝑐 = 𝑑 → (𝐵 +no 𝑐) = (𝐵 +no 𝑑))
3	1, 2	sseq12d 3942	. . . . . 6 ⊢ (𝑐 = 𝑑 → ((𝐴 +no 𝑐) ⊆ (𝐵 +no 𝑐) ↔ (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)))
4	3	imbi2d 343	. . . . 5 ⊢ (𝑐 = 𝑑 → ((𝐴 ⊆ 𝐵 → (𝐴 +no 𝑐) ⊆ (𝐵 +no 𝑐)) ↔ (𝐴 ⊆ 𝐵 → (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑))))
5	4	imbi2d 343	. . . 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 7432	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (𝐴 +no 𝑐) = (𝐴 +no 𝐶))
7		oveq2 7432	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (𝐵 +no 𝑐) = (𝐵 +no 𝐶))
8	6, 7	sseq12d 3942	. . . . . 6 ⊢ (𝑐 = 𝐶 → ((𝐴 +no 𝑐) ⊆ (𝐵 +no 𝑐) ↔ (𝐴 +no 𝐶) ⊆ (𝐵 +no 𝐶)))
9	8	imbi2d 343	. . . . 5 ⊢ (𝑐 = 𝐶 → ((𝐴 ⊆ 𝐵 → (𝐴 +no 𝑐) ⊆ (𝐵 +no 𝑐)) ↔ (𝐴 ⊆ 𝐵 → (𝐴 +no 𝐶) ⊆ (𝐵 +no 𝐶))))
10	9	imbi2d 343	. . . 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 3187	. . . . . 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 3187	. . . . . . 7 ⊢ (∀𝑑 ∈ 𝑐 (𝐴 ⊆ 𝐵 → (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) ↔ (𝐴 ⊆ 𝐵 → ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)))
13	12	imbi2i 339	. . . . . 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 278	. . . . 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 7432	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑑 = 𝑤 → (𝐴 +no 𝑑) = (𝐴 +no 𝑤))
16		oveq2 7432	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑑 = 𝑤 → (𝐵 +no 𝑑) = (𝐵 +no 𝑤))
17	15, 16	sseq12d 3942	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = 𝑤 → ((𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑) ↔ (𝐴 +no 𝑤) ⊆ (𝐵 +no 𝑤)))
18	17	rspccva 3575	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑) ∧ 𝑤 ∈ 𝑐) → (𝐴 +no 𝑤) ⊆ (𝐵 +no 𝑤))
19	18	ad4ant24 767	. . . . . . . . . . . . . . . 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 793	. . . . . . . . . . . . . . . . 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 7432	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑤 → (𝐵 +no 𝑦) = (𝐵 +no 𝑤))
22	21	eleq1d 2845	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑤 → ((𝐵 +no 𝑦) ∈ 𝑥 ↔ (𝐵 +no 𝑤) ∈ 𝑥))
23	22	rspccva 3575	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑦 ∈ 𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ 𝑤 ∈ 𝑐) → (𝐵 +no 𝑤) ∈ 𝑥)
24	20, 23	sylan 592	. . . . . . . . . . . . . . . 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 789	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ On)
26	25	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) → 𝐴 ∈ On)
27	26	adantr 486	. . . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . 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 795	. . . . . . . . . . . . . . . . . . 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 6390	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑐 ∈ On ∧ 𝑤 ∈ 𝑐) → 𝑤 ∈ On)
31	29, 30	sylan 592	. . . . . . . . . . . . . . . . . 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	28, 31	naddcld 8696	. . . . . . . . . . . . . . . . 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)
33		simplrl 789	. . . . . . . . . . . . . . . . 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)
34		ontr2 6414	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 +no 𝑤) ∈ On ∧ 𝑥 ∈ On) → (((𝐴 +no 𝑤) ⊆ (𝐵 +no 𝑤) ∧ (𝐵 +no 𝑤) ∈ 𝑥) → (𝐴 +no 𝑤) ∈ 𝑥))
35	32, 33, 34	syl2anc 596	. . . . . . . . . . . . . . . 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))
36	19, 24, 35	mp2and 712	. . . . . . . . . . . . . . 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)
37	36	ralrimiva 3154	. . . . . . . . . . . . . 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)
38		simpllr 788	. . . . . . . . . . . . . . 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)
39		simprrr 794	. . . . . . . . . . . . . . 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)
40		ssralv 3978	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑧 ∈ 𝐵 (𝑧 +no 𝑐) ∈ 𝑥 → ∀𝑧 ∈ 𝐴 (𝑧 +no 𝑐) ∈ 𝑥))
41	38, 39, 40	sylc 66	. . . . . . . . . . . . . 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)
42	37, 41	jca 521	. . . . . . . . . . . . 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))
43	42	expr 462	. . . . . . . . . . . 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)))
44	43	ss2rabdv 3999	. . . . . . . . . . 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)})
45		intss 4928	. . . . . . . . . . 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)})
46	44, 45	syl 18	. . . . . . . . . 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)})
47		simplll 787	. . . . . . . . . . 11 ⊢ ((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) → 𝑐 ∈ On)
48		naddov2 8695	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑐 ∈ On) → (𝐴 +no 𝑐) = ∩ {𝑥 ∈ On ∣ (∀𝑤 ∈ 𝑐 (𝐴 +no 𝑤) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝐴 (𝑧 +no 𝑐) ∈ 𝑥)})
49	26, 47, 48	syl2anc 596	. . . . . . . . . 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)})
50		simplrr 790	. . . . . . . . . . . 12 ⊢ (((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) → 𝐵 ∈ On)
51	50	adantr 486	. . . . . . . . . . 11 ⊢ ((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) → 𝐵 ∈ On)
52		naddov2 8695	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝑐 ∈ On) → (𝐵 +no 𝑐) = ∩ {𝑥 ∈ On ∣ (∀𝑦 ∈ 𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (𝑧 +no 𝑐) ∈ 𝑥)})
53	51, 47, 52	syl2anc 596	. . . . . . . . . 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)})
54	46, 49, 53	3sstr4d 3964	. . . . . . . . 9 ⊢ ((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) → (𝐴 +no 𝑐) ⊆ (𝐵 +no 𝑐))
55	54	exp31 425	. . . . . . . 8 ⊢ ((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → (𝐴 ⊆ 𝐵 → (∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑) → (𝐴 +no 𝑐) ⊆ (𝐵 +no 𝑐))))
56	55	a2d 30	. . . . . . 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))))
57	56	ex 418	. . . . . 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)))))
58	57	a2d 30	. . . . 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)))))
59	14, 58	biimtrid 245	. . . 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)))))
60	5, 10, 59	tfis3 7873	. . 3 ⊢ (𝐶 ∈ On → ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 +no 𝐶) ⊆ (𝐵 +no 𝐶))))
61	60	com12 33	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ∈ On → (𝐴 ⊆ 𝐵 → (𝐴 +no 𝐶) ⊆ (𝐵 +no 𝐶))))
62	61	3impia 1135	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 +no 𝐶) ⊆ (𝐵 +no 𝐶)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076  {crab 3412   ⊆ wss 3896  ∩ cint 4906  Oncon0 6365  (class class class)co 7424   +no cnadd 8681

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 5327  ax-pr 5391  ax-un 7755

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-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8005  df-2nd 8006  df-frecs 8306  df-nadd 8682

This theorem is used by:  naddel1  8704  nadd2rabex  44497