Theorem naddgeoa 44181

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 7426	. . 3 ⊢ (𝑎 = 𝑐 → (𝑎 +o 𝑏) = (𝑐 +o 𝑏))
2		oveq1 7426	. . 3 ⊢ (𝑎 = 𝑐 → (𝑎 +no 𝑏) = (𝑐 +no 𝑏))
3	1, 2	sseq12d 3971	. 2 ⊢ (𝑎 = 𝑐 → ((𝑎 +o 𝑏) ⊆ (𝑎 +no 𝑏) ↔ (𝑐 +o 𝑏) ⊆ (𝑐 +no 𝑏)))
4		oveq2 7427	. . 3 ⊢ (𝑏 = 𝑑 → (𝑐 +o 𝑏) = (𝑐 +o 𝑑))
5		oveq2 7427	. . 3 ⊢ (𝑏 = 𝑑 → (𝑐 +no 𝑏) = (𝑐 +no 𝑑))
6	4, 5	sseq12d 3971	. 2 ⊢ (𝑏 = 𝑑 → ((𝑐 +o 𝑏) ⊆ (𝑐 +no 𝑏) ↔ (𝑐 +o 𝑑) ⊆ (𝑐 +no 𝑑)))
7		oveq1 7426	. . 3 ⊢ (𝑎 = 𝑐 → (𝑎 +o 𝑑) = (𝑐 +o 𝑑))
8		oveq1 7426	. . 3 ⊢ (𝑎 = 𝑐 → (𝑎 +no 𝑑) = (𝑐 +no 𝑑))
9	7, 8	sseq12d 3971	. 2 ⊢ (𝑎 = 𝑐 → ((𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑) ↔ (𝑐 +o 𝑑) ⊆ (𝑐 +no 𝑑)))
10		oveq1 7426	. . 3 ⊢ (𝑎 = 𝐴 → (𝑎 +o 𝑏) = (𝐴 +o 𝑏))
11		oveq1 7426	. . 3 ⊢ (𝑎 = 𝐴 → (𝑎 +no 𝑏) = (𝐴 +no 𝑏))
12	10, 11	sseq12d 3971	. 2 ⊢ (𝑎 = 𝐴 → ((𝑎 +o 𝑏) ⊆ (𝑎 +no 𝑏) ↔ (𝐴 +o 𝑏) ⊆ (𝐴 +no 𝑏)))
13		oveq2 7427	. . 3 ⊢ (𝑏 = 𝐵 → (𝐴 +o 𝑏) = (𝐴 +o 𝐵))
14		oveq2 7427	. . 3 ⊢ (𝑏 = 𝐵 → (𝐴 +no 𝑏) = (𝐴 +no 𝐵))
15	13, 14	sseq12d 3971	. 2 ⊢ (𝑏 = 𝐵 → ((𝐴 +o 𝑏) ⊆ (𝐴 +no 𝑏) ↔ (𝐴 +o 𝐵) ⊆ (𝐴 +no 𝐵)))
16		simplll 787	. . . . . 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 788	. . . . . . 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 781	. . . . . . 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 521	. . . . . 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 8523	. . . . . 6 ⊢ ((𝑎 ∈ On ∧ (𝑏 ∈ On ∧ Lim 𝑏)) → (𝑎 +o 𝑏) = ∪ 𝑑 ∈ 𝑏 (𝑎 +o 𝑑))
21	16, 19, 20	syl2anc 596	. . . . 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 488	. . . . . 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 490	. . . . . . . . . . 11 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑 ∈ 𝑏) ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑)) → (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))
25		simpr 490	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → 𝑏 ∈ On)
26		onelss 6407	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ On → (𝑑 ∈ 𝑏 → 𝑑 ⊆ 𝑏))
27	25, 26	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (𝑑 ∈ 𝑏 → 𝑑 ⊆ 𝑏))
28	27	imp 412	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑 ∈ 𝑏) → 𝑑 ⊆ 𝑏)
29		simplr 781	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑 ∈ 𝑏) → 𝑏 ∈ On)
30		simpr 490	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑 ∈ 𝑏) → 𝑑 ∈ 𝑏)
31		onelon 6389	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ∈ On ∧ 𝑑 ∈ 𝑏) → 𝑑 ∈ On)
32	29, 30, 31	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑 ∈ 𝑏) → 𝑑 ∈ On)
33		simpll 779	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑 ∈ 𝑏) → 𝑎 ∈ On)
34		naddss2 8683	. . . . . . . . . . . . . 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 486	. . . . . . . . . . 11 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑 ∈ 𝑏) ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑)) → (𝑎 +no 𝑑) ⊆ (𝑎 +no 𝑏))
38	24, 37	sstrd 3948	. . . . . . . . . 10 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑 ∈ 𝑏) ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑)) → (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑏))
39	38	ex 418	. . . . . . . . 9 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑 ∈ 𝑏) → ((𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑) → (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑏)))
40	39	ralimdva 3179	. . . . . . . 8 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∀𝑑 ∈ 𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑) → ∀𝑑 ∈ 𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑏)))
41	40	imp 412	. . . . . . 7 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ ∀𝑑 ∈ 𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑)) → ∀𝑑 ∈ 𝑏 (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑏))
42		iunss 5011	. . . . . . 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 608	. . . . 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 3972	. . . 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 425	. . 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 7849	. . . . . . 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 407	. . . . . 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 6374	. . . . . 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 3961	. . . . . . . 8 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑏 = ∅) → 𝑎 ⊆ 𝑎)
56		simpr 490	. . . . . . . . . 10 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑏 = ∅) → 𝑏 = ∅)
57	56	oveq2d 7435	. . . . . . . . 9 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑏 = ∅) → (𝑎 +o 𝑏) = (𝑎 +o ∅))
58		simpll 779	. . . . . . . . . 10 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑏 = ∅) → 𝑎 ∈ On)
59		oa0 8507	. . . . . . . . . 10 ⊢ (𝑎 ∈ On → (𝑎 +o ∅) = 𝑎)
60	58, 59	syl 18	. . . . . . . . 9 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑏 = ∅) → (𝑎 +o ∅) = 𝑎)
61	57, 60	eqtrd 2800	. . . . . . . 8 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑏 = ∅) → (𝑎 +o 𝑏) = 𝑎)
62	56	oveq2d 7435	. . . . . . . . 9 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑏 = ∅) → (𝑎 +no 𝑏) = (𝑎 +no ∅))
63		naddrid 8676	. . . . . . . . . 10 ⊢ (𝑎 ∈ On → (𝑎 +no ∅) = 𝑎)
64	58, 63	syl 18	. . . . . . . . 9 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑏 = ∅) → (𝑎 +no ∅) = 𝑎)
65	62, 64	eqtrd 2800	. . . . . . . 8 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑏 = ∅) → (𝑎 +no 𝑏) = 𝑎)
66	55, 61, 65	3sstr4d 3993	. . . . . . 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 418	. . . . 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 3461	. . . . . . . . . . 11 ⊢ 𝑑 ∈ V
70	69	sucid 6449	. . . . . . . . . 10 ⊢ 𝑑 ∈ suc 𝑑
71		simpr 490	. . . . . . . . . 10 ⊢ ((𝑑 ∈ On ∧ 𝑏 = suc 𝑑) → 𝑏 = suc 𝑑)
72	70, 71	eleqtrrid 2872	. . . . . . . . 9 ⊢ ((𝑑 ∈ On ∧ 𝑏 = suc 𝑑) → 𝑑 ∈ 𝑏)
73	72, 71	jca 521	. . . . . . . 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 3177	. . . . . 6 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (∃𝑑 ∈ On 𝑏 = suc 𝑑 → ∃𝑑 ∈ 𝑏 𝑏 = suc 𝑑))
76		r19.29r 3131	. . . . . . . . 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 785	. . . . . . . . . . . 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 521	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑 ∈ 𝑏) → (𝑎 ∈ On ∧ 𝑑 ∈ On))
79		oacl 8526	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ On ∧ 𝑑 ∈ On) → (𝑎 +o 𝑑) ∈ On)
80		eloni 6374	. . . . . . . . . . . . . . . 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 8669	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ On ∧ 𝑑 ∈ On) → (𝑎 +no 𝑑) ∈ On)
83		eloni 6374	. . . . . . . . . . . . . . . 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 521	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ On ∧ 𝑑 ∈ On) → (Ord (𝑎 +o 𝑑) ∧ Ord (𝑎 +no 𝑑)))
86		ordsucsssuc 7825	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . 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 783	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑 ∈ 𝑏) ∧ (𝑏 = suc 𝑑 ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → 𝑏 = suc 𝑑)
91	90	oveq2d 7435	. . . . . . . . . . . 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 486	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑 ∈ 𝑏) ∧ (𝑏 = suc 𝑑 ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → (𝑎 ∈ On ∧ 𝑑 ∈ On))
93		oasuc 8515	. . . . . . . . . . . . 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 2800	. . . . . . . . . . 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 7435	. . . . . . . . . . . 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 787	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑 ∈ 𝑏) ∧ (𝑏 = suc 𝑑 ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → 𝑎 ∈ On)
98	31	ad4ant23 766	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ On ∧ 𝑏 ∈ On) ∧ 𝑑 ∈ 𝑏) ∧ (𝑏 = suc 𝑑 ∧ (𝑎 +o 𝑑) ⊆ (𝑎 +no 𝑑))) → 𝑑 ∈ On)
99		naddsuc2 8694	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ On ∧ 𝑑 ∈ On) → (𝑎 +no suc 𝑑) = suc (𝑎 +no 𝑑))
100	97, 98, 99	syl2anc 596	. . . . . . . . . . . 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 2800	. . . . . . . . . . 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 3993	. . . . . . . . . 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 3170	. . . . . . . . 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 421	. . . . . . 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 873	. . . 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 8661	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) ⊆ (𝐴 +no 𝐵))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091   ⊆ wss 3906  ∅c0 4286  ∪ ciun 4958  Ord word 6363  Oncon0 6364  Lim wlim 6365  suc csuc 6366  (class class class)co 7419   +_o coa 8456   +no cnadd 8657

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-oadd 8463  df-nadd 8658

This theorem is used by:  naddwordnexlem4  44188