Theorem naddssim 8630

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 7365	. . . . . . 7 ⊢ (𝑐 = 𝑑 → (𝐴 +no 𝑐) = (𝐴 +no 𝑑))
2		oveq2 7365	. . . . . . 7 ⊢ (𝑐 = 𝑑 → (𝐵 +no 𝑐) = (𝐵 +no 𝑑))
3	1, 2	sseq12d 3977	. . . . . 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 7365	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (𝐴 +no 𝑐) = (𝐴 +no 𝐶))
7		oveq2 7365	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (𝐵 +no 𝑐) = (𝐵 +no 𝐶))
8	6, 7	sseq12d 3977	. . . . . 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 3176	. . . . . 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 3176	. . . . . . 7 ⊢ (∀𝑑 ∈ 𝑐 (𝐴 ⊆ 𝐵 → (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) ↔ (𝐴 ⊆ 𝐵 → ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)))
13	12	imbi2i 335	. . . . . 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 274	. . . . 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 7365	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑑 = 𝑤 → (𝐴 +no 𝑑) = (𝐴 +no 𝑤))
16		oveq2 7365	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑑 = 𝑤 → (𝐵 +no 𝑑) = (𝐵 +no 𝑤))
17	15, 16	sseq12d 3977	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑑 = 𝑤 → ((𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑) ↔ (𝐴 +no 𝑤) ⊆ (𝐵 +no 𝑤)))
18	17	rspccva 3580	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑) ∧ 𝑤 ∈ 𝑐) → (𝐴 +no 𝑤) ⊆ (𝐵 +no 𝑤))
19	18	ad4ant24 752	. . . . . . . . . . . . . . . 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 779	. . . . . . . . . . . . . . . . 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 7365	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑤 → (𝐵 +no 𝑦) = (𝐵 +no 𝑤))
22	21	eleq1d 2822	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑤 → ((𝐵 +no 𝑦) ∈ 𝑥 ↔ (𝐵 +no 𝑤) ∈ 𝑥))
23	22	rspccva 3580	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑦 ∈ 𝑐 (𝐵 +no 𝑦) ∈ 𝑥 ∧ 𝑤 ∈ 𝑐) → (𝐵 +no 𝑤) ∈ 𝑥)
24	20, 23	sylan 580	. . . . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ On)
26	25	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) → 𝐴 ∈ On)
27	26	adantr 481	. . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . 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 781	. . . . . . . . . . . . . . . . . . 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 6342	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑐 ∈ On ∧ 𝑤 ∈ 𝑐) → 𝑤 ∈ On)
31	29, 30	sylan 580	. . . . . . . . . . . . . . . . . 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 8623	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ On ∧ 𝑤 ∈ On) → (𝐴 +no 𝑤) ∈ On)
33	28, 31, 32	syl2anc 584	. . . . . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . . . 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 6364	. . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . 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 697	. . . . . . . . . . . . . . 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 3143	. . . . . . . . . . . . . 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 774	. . . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . 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 4010	. . . . . . . . . . . . . . 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 512	. . . . . . . . . . . . 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 457	. . . . . . . . . . . 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 4033	. . . . . . . . . . 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 4930	. . . . . . . . . . 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 773	. . . . . . . . . . 11 ⊢ ((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) → 𝑐 ∈ On)
49		naddov2 8625	. . . . . . . . . . 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 584	. . . . . . . . . 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 776	. . . . . . . . . . . 12 ⊢ (((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) → 𝐵 ∈ On)
52	51	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝑐 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑑 ∈ 𝑐 (𝐴 +no 𝑑) ⊆ (𝐵 +no 𝑑)) → 𝐵 ∈ On)
53		naddov2 8625	. . . . . . . . . . 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 584	. . . . . . . . . 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 3991	. . . . . . . . 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 420	. . . . . . . 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 413	. . . . . 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 241	. . . 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 7794	. . 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 1117	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 +no 𝐶) ⊆ (𝐵 +no 𝐶)))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3064  {crab 3407   ⊆ wss 3910  ∩ cint 4907  Oncon0 6317  (class class class)co 7357   +no cnadd 8611

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-frecs 8212  df-nadd 8612

This theorem is referenced by:  naddel1  8632