Theorem naddgeoa 44141

Description: Natural addition results in a value greater than or equal than that of ordinal addition. (Contributed by RP, 1-Jan-2025.)

Assertion

Ref	Expression
naddgeoa	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ⊆ (𝐴 +no 𝐵))

Proof of Theorem naddgeoa

Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7417	. . 3 ⊢ (𝑎 = 𝑐 → (𝑎 +o 𝑏) = (𝑐 +o 𝑏))
2		oveq1 7417	. . 3 ⊢ (𝑎 = 𝑐 → (𝑎 +no 𝑏) = (𝑐 +no 𝑏))
3	1, 2	sseq12d 3970	. 2 ⊢ (𝑎 = 𝑐 → ((𝑎 +o 𝑏) ⊆ (𝑎 +no 𝑏) ↔ (𝑐 +o 𝑏) ⊆ (𝑐 +no 𝑏)))
4		oveq2 7418	. . 3 ⊢ (𝑏 = 𝑑 → (𝑐 +o 𝑏) = (𝑐 +o 𝑑))
5		oveq2 7418	. . 3 ⊢ (𝑏 = 𝑑 → (𝑐 +no 𝑏) = (𝑐 +no 𝑑))
6	4, 5	sseq12d 3970	. 2 ⊢ (𝑏 = 𝑑 → ((𝑐 +o 𝑏) ⊆ (𝑐 +no 𝑏) ↔ (𝑐 +o 𝑑) ⊆ (𝑐 +no 𝑑)))
7		oveq1 7417	. . 3 ⊢ (𝑎 = 𝑐 → (𝑎 +o 𝑑) = (𝑐 +o 𝑑))
8		oveq1 7417	. . 3 ⊢ (𝑎 = 𝑐 → (𝑎 +no 𝑑) = (𝑐 +no 𝑑))
9	7, 8	sseq12d 3970	. 2 ⊢ (𝑎 = 𝑐 → ((𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑) ↔ (𝑐 +o 𝑑) ⊆ (𝑐 +no 𝑑)))
10		oveq1 7417	. . 3 ⊢ (𝑎 = 𝐴 → (𝑎 +o 𝑏) = (𝐴 +o 𝑏))
11		oveq1 7417	. . 3 ⊢ (𝑎 = 𝐴 → (𝑎 +no 𝑏) = (𝐴 +no 𝑏))
12	10, 11	sseq12d 3970	. 2 ⊢ (𝑎 = 𝐴 → ((𝑎 +o 𝑏) ⊆ (𝑎 +no 𝑏) ↔ (𝐴 +o 𝑏) ⊆ (𝐴 +no 𝑏)))
13		oveq2 7418	. . 3 ⊢ (𝑏 = 𝐵 → (𝐴 +o 𝑏) = (𝐴 +o 𝐵))
14		oveq2 7418	. . 3 ⊢ (𝑏 = 𝐵 → (𝐴 +no 𝑏) = (𝐴 +no 𝐵))
15	13, 14	sseq12d 3970	. 2 ⊢ (𝑏 = 𝐵 → ((𝐴 +o 𝑏) ⊆ (𝐴 +no 𝑏) ↔ (𝐴 +o 𝐵) ⊆ (𝐴 +no 𝐵)))
16		simplll 786	. . . . . 6 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ Lim 𝑏) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 +o 𝑑) ⊆ (𝑐 +no 𝑑) ∧ ∀𝑐 ∈ 𝑎 (𝑐 +o 𝑏) ⊆ (𝑐 +no 𝑏) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → 𝑎 ∈ On)
17		simpllr 787	. . . . . . 7 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ Lim 𝑏) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 +o 𝑑) ⊆ (𝑐 +no 𝑑) ∧ ∀𝑐 ∈ 𝑎 (𝑐 +o 𝑏) ⊆ (𝑐 +no 𝑏) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → 𝑏 ∈ On)
18		simplr 780	. . . . . . 7 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ Lim 𝑏) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 +o 𝑑) ⊆ (𝑐 +no 𝑑) ∧ ∀𝑐 ∈ 𝑎 (𝑐 +o 𝑏) ⊆ (𝑐 +no 𝑏) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → Lim 𝑏)
19	17, 18	jca 520	. . . . . 6 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ Lim 𝑏) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 +o 𝑑) ⊆ (𝑐 +no 𝑑) ∧ ∀𝑐 ∈ 𝑎 (𝑐 +o 𝑏) ⊆ (𝑐 +no 𝑏) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → (𝑏 ∈ On ∧ Lim 𝑏))
20		oalim 8513	. . . . . 6 ⊢ ((𝑎 ∈ On ∧ (𝑏 ∈ On ∧ Lim 𝑏)) → (𝑎 +o 𝑏) = ∪ 𝑑 ∈ 𝑏 (𝑎 +o 𝑑))
21	16, 19, 20	syl2anc 595	. . . . 5 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ Lim 𝑏) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 +o 𝑑) ⊆ (𝑐 +no 𝑑) ∧ ∀𝑐 ∈ 𝑎 (𝑐 +o 𝑏) ⊆ (𝑐 +no 𝑏) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → (𝑎 +o 𝑏) = ∪ 𝑑 ∈ 𝑏 (𝑎 +o 𝑑))
22		simpl 487	. . . . . 6 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ Lim 𝑏) → (𝑎 ∈ On ∧ 𝑏 ∈ On))
23		simp3 1156	. . . . . 6 ⊢ ((∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 +o 𝑑) ⊆ (𝑐 +no 𝑑) ∧ ∀𝑐 ∈ 𝑎 (𝑐 +o 𝑏) ⊆ (𝑐 +no 𝑏) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑)) → ∀𝑑 ∈ 𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))
24		simpr 489	. . . . . . . . . . 11 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑 ∈ 𝑏) ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑)) → (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))
25		simpr 489	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → 𝑏 ∈ On)
26		onelss 6403	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ On → (𝑑 ∈ 𝑏 → 𝑑 ⊆ 𝑏))
27	25, 26	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑑 ∈ 𝑏 → 𝑑 ⊆ 𝑏))
28	27	imp 411	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑 ∈ 𝑏) → 𝑑 ⊆ 𝑏)
29		simplr 780	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑 ∈ 𝑏) → 𝑏 ∈ On)
30		simpr 489	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑 ∈ 𝑏) → 𝑑 ∈ 𝑏)
31		onelon 6385	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ∈ On ∧ 𝑑 ∈ 𝑏) → 𝑑 ∈ On)
32	29, 30, 31	syl2anc 595	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑 ∈ 𝑏) → 𝑑 ∈ On)
33		simpll 778	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑 ∈ 𝑏) → 𝑎 ∈ On)
34		naddss2 8673	. . . . . . . . . . . . . 14 ⊢ ((𝑑 ∈ On ∧ 𝑏 ∈ On ∧ 𝑎 ∈ On) → (𝑑 ⊆ 𝑏 ↔ (𝑎 +no 𝑑) ⊆ (𝑎 +no 𝑏)))
35	32, 29, 33, 34	syl3anc 1398	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑 ∈ 𝑏) → (𝑑 ⊆ 𝑏 ↔ (𝑎 +no 𝑑) ⊆ (𝑎 +no 𝑏)))
36	28, 35	mpbid 235	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑 ∈ 𝑏) → (𝑎 +no 𝑑) ⊆ (𝑎 +no 𝑏))
37	36	adantr 485	. . . . . . . . . . 11 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑 ∈ 𝑏) ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑)) → (𝑎 +no 𝑑) ⊆ (𝑎 +no 𝑏))
38	24, 37	sstrd 3947	. . . . . . . . . 10 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑 ∈ 𝑏) ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑)) → (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑏))
39	38	ex 417	. . . . . . . . 9 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑 ∈ 𝑏) → ((𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑) → (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑏)))
40	39	ralimdva 3177	. . . . . . . 8 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑑 ∈ 𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑) → ∀𝑑 ∈ 𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑏)))
41	40	imp 411	. . . . . . 7 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑)) → ∀𝑑 ∈ 𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑏))
42		iunss 5009	. . . . . . 7 ⊢ (∪ 𝑑 ∈ 𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑏) ↔ ∀𝑑 ∈ 𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑏))
43	41, 42	sylibr 237	. . . . . 6 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑)) → ∪ 𝑑 ∈ 𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑏))
44	22, 23, 43	syl2an 607	. . . . 5 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ Lim 𝑏) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 +o 𝑑) ⊆ (𝑐 +no 𝑑) ∧ ∀𝑐 ∈ 𝑎 (𝑐 +o 𝑏) ⊆ (𝑐 +no 𝑏) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → ∪ 𝑑 ∈ 𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑏))
45	21, 44	eqsstrd 3971	. . . 4 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ Lim 𝑏) ∧ (∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 +o 𝑑) ⊆ (𝑐 +no 𝑑) ∧ ∀𝑐 ∈ 𝑎 (𝑐 +o 𝑏) ⊆ (𝑐 +no 𝑏) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → (𝑎 +o 𝑏) ⊆ (𝑎 +no 𝑏))
46	45	exp31 424	. . 3 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (Lim 𝑏 → ((∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 +o 𝑑) ⊆ (𝑐 +no 𝑑) ∧ ∀𝑐 ∈ 𝑎 (𝑐 +o 𝑏) ⊆ (𝑐 +no 𝑏) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑)) → (𝑎 +o 𝑏) ⊆ (𝑎 +no 𝑏))))
47		dflim3 7839	. . . . . . 7 ⊢ (Lim 𝑏 ↔ (Ord 𝑏 ∧ ¬ (𝑏 = ∅ ∨ ∃𝑑 ∈ On 𝑏 = suc 𝑑)))
48	47	notbii 323	. . . . . 6 ⊢ (¬ Lim 𝑏 ↔ ¬ (Ord 𝑏 ∧ ¬ (𝑏 = ∅ ∨ ∃𝑑 ∈ On 𝑏 = suc 𝑑)))
49		iman 406	. . . . . 6 ⊢ ((Ord 𝑏 → (𝑏 = ∅ ∨ ∃𝑑 ∈ On 𝑏 = suc 𝑑)) ↔ ¬ (Ord 𝑏 ∧ ¬ (𝑏 = ∅ ∨ ∃𝑑 ∈ On 𝑏 = suc 𝑑)))
50	48, 49	bitr4i 281	. . . . 5 ⊢ (¬ Lim 𝑏 ↔ (Ord 𝑏 → (𝑏 = ∅ ∨ ∃𝑑 ∈ On 𝑏 = suc 𝑑)))
51		eloni 6370	. . . . . 6 ⊢ (𝑏 ∈ On → Ord 𝑏)
52		pm5.5 364	. . . . . 6 ⊢ (Ord 𝑏 → ((Ord 𝑏 → (𝑏 = ∅ ∨ ∃𝑑 ∈ On 𝑏 = suc 𝑑)) ↔ (𝑏 = ∅ ∨ ∃𝑑 ∈ On 𝑏 = suc 𝑑)))
53	25, 51, 52	3syl 19	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((Ord 𝑏 → (𝑏 = ∅ ∨ ∃𝑑 ∈ On 𝑏 = suc 𝑑)) ↔ (𝑏 = ∅ ∨ ∃𝑑 ∈ On 𝑏 = suc 𝑑)))
54	50, 53	bitrid 286	. . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (¬ Lim 𝑏 ↔ (𝑏 = ∅ ∨ ∃𝑑 ∈ On 𝑏 = suc 𝑑)))
55		ssidd 3960	. . . . . . . 8 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑏 = ∅) → 𝑎 ⊆ 𝑎)
56		simpr 489	. . . . . . . . . 10 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑏 = ∅) → 𝑏 = ∅)
57	56	oveq2d 7426	. . . . . . . . 9 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑏 = ∅) → (𝑎 +o 𝑏) = (𝑎 +o ∅))
58		simpll 778	. . . . . . . . . 10 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑏 = ∅) → 𝑎 ∈ On)
59		oa0 8497	. . . . . . . . . 10 ⊢ (𝑎 ∈ On → (𝑎 +o ∅) = 𝑎)
60	58, 59	syl 18	. . . . . . . . 9 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑏 = ∅) → (𝑎 +o ∅) = 𝑎)
61	57, 60	eqtrd 2798	. . . . . . . 8 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑏 = ∅) → (𝑎 +o 𝑏) = 𝑎)
62	56	oveq2d 7426	. . . . . . . . 9 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑏 = ∅) → (𝑎 +no 𝑏) = (𝑎 +no ∅))
63		naddrid 8666	. . . . . . . . . 10 ⊢ (𝑎 ∈ On → (𝑎 +no ∅) = 𝑎)
64	58, 63	syl 18	. . . . . . . . 9 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑏 = ∅) → (𝑎 +no ∅) = 𝑎)
65	62, 64	eqtrd 2798	. . . . . . . 8 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑏 = ∅) → (𝑎 +no 𝑏) = 𝑎)
66	55, 61, 65	3sstr4d 3992	. . . . . . 7 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑏 = ∅) → (𝑎 +o 𝑏) ⊆ (𝑎 +no 𝑏))
67	66	a1d 26	. . . . . 6 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑏 = ∅) → ((∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 +o 𝑑) ⊆ (𝑐 +no 𝑑) ∧ ∀𝑐 ∈ 𝑎 (𝑐 +o 𝑏) ⊆ (𝑐 +no 𝑏) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑)) → (𝑎 +o 𝑏) ⊆ (𝑎 +no 𝑏)))
68	67	ex 417	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑏 = ∅ → ((∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 +o 𝑑) ⊆ (𝑐 +no 𝑑) ∧ ∀𝑐 ∈ 𝑎 (𝑐 +o 𝑏) ⊆ (𝑐 +no 𝑏) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑)) → (𝑎 +o 𝑏) ⊆ (𝑎 +no 𝑏))))
69		vex 3459	. . . . . . . . . . 11 ⊢ 𝑑 ∈ V
70	69	sucid 6445	. . . . . . . . . 10 ⊢ 𝑑 ∈ suc 𝑑
71		simpr 489	. . . . . . . . . 10 ⊢ ((𝑑 ∈ On ∧ 𝑏 = suc 𝑑) → 𝑏 = suc 𝑑)
72	70, 71	eleqtrrid 2870	. . . . . . . . 9 ⊢ ((𝑑 ∈ On ∧ 𝑏 = suc 𝑑) → 𝑑 ∈ 𝑏)
73	72, 71	jca 520	. . . . . . . 8 ⊢ ((𝑑 ∈ On ∧ 𝑏 = suc 𝑑) → (𝑑 ∈ 𝑏 ∧ 𝑏 = suc 𝑑))
74	73	a1i 11	. . . . . . 7 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝑑 ∈ On ∧ 𝑏 = suc 𝑑) → (𝑑 ∈ 𝑏 ∧ 𝑏 = suc 𝑑)))
75	74	reximdv2 3175	. . . . . 6 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∃𝑑 ∈ On 𝑏 = suc 𝑑 → ∃𝑑 ∈ 𝑏 𝑏 = suc 𝑑))
76		r19.29r 3129	. . . . . . . . 9 ⊢ ((∃𝑑 ∈ 𝑏 𝑏 = suc 𝑑 ∧ ∀𝑑 ∈ 𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑)) → ∃𝑑 ∈ 𝑏 (𝑏 = suc 𝑑 ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑)))
77		simprr 784	. . . . . . . . . . . 12 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑 ∈ 𝑏) ∧ (𝑏 = suc 𝑑 ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))
78	33, 32	jca 520	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑 ∈ 𝑏) → (𝑎 ∈ On ∧ 𝑑 ∈ On))
79		oacl 8516	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ On ∧ 𝑑 ∈ On) → (𝑎 +o 𝑑) ∈ On)
80		eloni 6370	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 +o 𝑑) ∈ On → Ord (𝑎 +o 𝑑))
81	79, 80	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ On ∧ 𝑑 ∈ On) → Ord (𝑎 +o 𝑑))
82		naddcl 8659	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ On ∧ 𝑑 ∈ On) → (𝑎 +no 𝑑) ∈ On)
83		eloni 6370	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 +no 𝑑) ∈ On → Ord (𝑎 +no 𝑑))
84	82, 83	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ On ∧ 𝑑 ∈ On) → Ord (𝑎 +no 𝑑))
85	81, 84	jca 520	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ On ∧ 𝑑 ∈ On) → (Ord (𝑎 +o 𝑑) ∧ Ord (𝑎 +no 𝑑)))
86		ordsucsssuc 7815	. . . . . . . . . . . . . 14 ⊢ ((Ord (𝑎 +o 𝑑) ∧ Ord (𝑎 +no 𝑑)) → ((𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑) ↔ suc (𝑎 +o 𝑑) ⊆ suc (𝑎 +no 𝑑)))
87	78, 85, 86	3syl 19	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑 ∈ 𝑏) → ((𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑) ↔ suc (𝑎 +o 𝑑) ⊆ suc (𝑎 +no 𝑑)))
88	87	adantr 485	. . . . . . . . . . . 12 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑 ∈ 𝑏) ∧ (𝑏 = suc 𝑑 ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → ((𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑) ↔ suc (𝑎 +o 𝑑) ⊆ suc (𝑎 +no 𝑑)))
89	77, 88	mpbid 235	. . . . . . . . . . 11 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑 ∈ 𝑏) ∧ (𝑏 = suc 𝑑 ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → suc (𝑎 +o 𝑑) ⊆ suc (𝑎 +no 𝑑))
90		simprl 782	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑 ∈ 𝑏) ∧ (𝑏 = suc 𝑑 ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → 𝑏 = suc 𝑑)
91	90	oveq2d 7426	. . . . . . . . . . . 12 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑 ∈ 𝑏) ∧ (𝑏 = suc 𝑑 ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → (𝑎 +o 𝑏) = (𝑎 +o suc 𝑑))
92	78	adantr 485	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑 ∈ 𝑏) ∧ (𝑏 = suc 𝑑 ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → (𝑎 ∈ On ∧ 𝑑 ∈ On))
93		oasuc 8505	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ On ∧ 𝑑 ∈ On) → (𝑎 +o suc 𝑑) = suc (𝑎 +o 𝑑))
94	92, 93	syl 18	. . . . . . . . . . . 12 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑 ∈ 𝑏) ∧ (𝑏 = suc 𝑑 ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → (𝑎 +o suc 𝑑) = suc (𝑎 +o 𝑑))
95	91, 94	eqtrd 2798	. . . . . . . . . . 11 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑 ∈ 𝑏) ∧ (𝑏 = suc 𝑑 ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → (𝑎 +o 𝑏) = suc (𝑎 +o 𝑑))
96	90	oveq2d 7426	. . . . . . . . . . . 12 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑 ∈ 𝑏) ∧ (𝑏 = suc 𝑑 ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → (𝑎 +no 𝑏) = (𝑎 +no suc 𝑑))
97		simplll 786	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑 ∈ 𝑏) ∧ (𝑏 = suc 𝑑 ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → 𝑎 ∈ On)
98	31	ad4ant23 765	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑 ∈ 𝑏) ∧ (𝑏 = suc 𝑑 ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → 𝑑 ∈ On)
99		naddsuc2 8684	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ On ∧ 𝑑 ∈ On) → (𝑎 +no suc 𝑑) = suc (𝑎 +no 𝑑))
100	97, 98, 99	syl2anc 595	. . . . . . . . . . . 12 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑 ∈ 𝑏) ∧ (𝑏 = suc 𝑑 ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → (𝑎 +no suc 𝑑) = suc (𝑎 +no 𝑑))
101	96, 100	eqtrd 2798	. . . . . . . . . . 11 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑 ∈ 𝑏) ∧ (𝑏 = suc 𝑑 ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → (𝑎 +no 𝑏) = suc (𝑎 +no 𝑑))
102	89, 95, 101	3sstr4d 3992	. . . . . . . . . 10 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑 ∈ 𝑏) ∧ (𝑏 = suc 𝑑 ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → (𝑎 +o 𝑏) ⊆ (𝑎 +no 𝑏))
103	102	rexlimdva2 3168	. . . . . . . . 9 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∃𝑑 ∈ 𝑏 (𝑏 = suc 𝑑 ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑)) → (𝑎 +o 𝑏) ⊆ (𝑎 +no 𝑏)))
104	76, 103	syl5 35	. . . . . . . 8 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((∃𝑑 ∈ 𝑏 𝑏 = suc 𝑑 ∧ ∀𝑑 ∈ 𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑)) → (𝑎 +o 𝑏) ⊆ (𝑎 +no 𝑏)))
105	104	expd 420	. . . . . . 7 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∃𝑑 ∈ 𝑏 𝑏 = suc 𝑑 → (∀𝑑 ∈ 𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑) → (𝑎 +o 𝑏) ⊆ (𝑎 +no 𝑏))))
106	23, 105	syl7 75	. . . . . 6 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∃𝑑 ∈ 𝑏 𝑏 = suc 𝑑 → ((∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 +o 𝑑) ⊆ (𝑐 +no 𝑑) ∧ ∀𝑐 ∈ 𝑎 (𝑐 +o 𝑏) ⊆ (𝑐 +no 𝑏) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑)) → (𝑎 +o 𝑏) ⊆ (𝑎 +no 𝑏))))
107	75, 106	syld 48	. . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∃𝑑 ∈ On 𝑏 = suc 𝑑 → ((∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 +o 𝑑) ⊆ (𝑐 +no 𝑑) ∧ ∀𝑐 ∈ 𝑎 (𝑐 +o 𝑏) ⊆ (𝑐 +no 𝑏) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑)) → (𝑎 +o 𝑏) ⊆ (𝑎 +no 𝑏))))
108	68, 107	jaod 872	. . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝑏 = ∅ ∨ ∃𝑑 ∈ On 𝑏 = suc 𝑑) → ((∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 +o 𝑑) ⊆ (𝑐 +no 𝑑) ∧ ∀𝑐 ∈ 𝑎 (𝑐 +o 𝑏) ⊆ (𝑐 +no 𝑏) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑)) → (𝑎 +o 𝑏) ⊆ (𝑎 +no 𝑏))))
109	54, 108	sylbid 243	. . 3 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (¬ Lim 𝑏 → ((∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 +o 𝑑) ⊆ (𝑐 +no 𝑑) ∧ ∀𝑐 ∈ 𝑎 (𝑐 +o 𝑏) ⊆ (𝑐 +no 𝑏) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑)) → (𝑎 +o 𝑏) ⊆ (𝑎 +no 𝑏))))
110	46, 109	pm2.61d 181	. 2 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((∀𝑐 ∈ 𝑎 ∀𝑑 ∈ 𝑏 (𝑐 +o 𝑑) ⊆ (𝑐 +no 𝑑) ∧ ∀𝑐 ∈ 𝑎 (𝑐 +o 𝑏) ⊆ (𝑐 +no 𝑏) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑)) → (𝑎 +o 𝑏) ⊆ (𝑎 +no 𝑏)))
111	3, 6, 9, 12, 15, 110	on2ind 8651	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ⊆ (𝐴 +no 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089   ⊆ wss 3905  ∅c0 4286  ∪ ciun 4956  Ord word 6359  Oncon0 6360  Lim wlim 6361  suc csuc 6362  (class class class)co 7410   +_o coa 8446   +no cnadd 8647

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-oadd 8453  df-nadd 8648

This theorem is referenced by:  naddwordnexlem4  44148