Theorem naddssim 8671

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 7419	. . . . . . 7 ⊢ (𝑐 = 𝑑 → (𝐴 +no 𝑐) = (𝐴 +no 𝑑))
2		oveq2 7419	. . . . . . 7 ⊢ (𝑐 = 𝑑 → (𝐵 +no 𝑐) = (𝐵 +no 𝑑))
3	1, 2	sseq12d 3978	. . . . . 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 7419	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (𝐴 +no 𝑐) = (𝐴 +no 𝐶))
7		oveq2 7419	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (𝐵 +no 𝑐) = (𝐵 +no 𝐶))
8	6, 7	sseq12d 3978	. . . . . 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 3196	. . . . . 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 3196	. . . . . . 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 7419	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑑 = 𝑤 → (𝐴 +no 𝑑) = (𝐴 +no 𝑤))
16		oveq2 7419	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑑 = 𝑤 → (𝐵 +no 𝑑) = (𝐵 +no 𝑤))
17	15, 16	sseq12d 3978	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = 𝑤 → ((𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑) ↔ (𝐴 +no 𝑤) ⊆ (𝐵 +no 𝑤)))
18	17	rspccva 3589	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑) ∧ 𝑤 ∈ 𝑐) → (𝐴 +no 𝑤) ⊆ (𝐵 +no 𝑤))
19	18	ad4ant24 766	. . . . . . . . . . . . . . . 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 792	. . . . . . . . . . . . . . . . 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 7419	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑤 → (𝐵 +no 𝑦) = (𝐵 +no 𝑤))
22	21	eleq1d 2854	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑤 → ((𝐵 +no 𝑦) ∈ 𝑥 ↔ (𝐵 +no 𝑤) ∈ 𝑥))
23	22	rspccva 3589	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑦 ∈ 𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ 𝑤 ∈ 𝑐) → (𝐵 +no 𝑤) ∈ 𝑥)
24	20, 23	sylan 591	. . . . . . . . . . . . . . . 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 788	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ On)
26	25	adantr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) → 𝐴 ∈ On)
27	26	adantr 485	. . . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . 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 794	. . . . . . . . . . . . . . . . . . 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 6386	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑐 ∈ On ∧ 𝑤 ∈ 𝑐) → 𝑤 ∈ On)
31	29, 30	sylan 591	. . . . . . . . . . . . . . . . . 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 8665	. . . . . . . . . . . . . . . . 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 788	. . . . . . . . . . . . . . . . 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 6410	. . . . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . . . 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 711	. . . . . . . . . . . . . . 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 3163	. . . . . . . . . . . . . 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 787	. . . . . . . . . . . . . . 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 793	. . . . . . . . . . . . . . 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 4014	. . . . . . . . . . . . . . 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 520	. . . . . . . . . . . . 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 461	. . . . . . . . . . . 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 4037	. . . . . . . . . . 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 4938	. . . . . . . . . . 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 786	. . . . . . . . . . 11 ⊢ ((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) → 𝑐 ∈ On)
48		naddov2 8664	. . . . . . . . . . 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 595	. . . . . . . . . 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 789	. . . . . . . . . . . 12 ⊢ (((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) → 𝐵 ∈ On)
51	50	adantr 485	. . . . . . . . . . 11 ⊢ ((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) → 𝐵 ∈ On)
52		naddov2 8664	. . . . . . . . . . 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 595	. . . . . . . . . 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 4000	. . . . . . . . 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 424	. . . . . . . 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 417	. . . . . 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 7853	. . 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 1133	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 +no 𝐶) ⊆ (𝐵 +no 𝐶)))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∀wral 3085  {crab 3423   ⊆ wss 3913  ∩ cint 4916  Oncon0 6361  (class class class)co 7411   +no cnadd 8650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7985  df-2nd 7986  df-frecs 8277  df-nadd 8651

This theorem is referenced by:  naddel1  8673  nadd2rabex  44004