Theorem naddasslem1 8661

Description: Lemma for naddass 8663. Expand out the expression for natural addition of three arguments. (Contributed by Scott Fenton, 20-Jan-2025.)

Assertion

Ref	Expression
naddasslem1	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 +no 𝐵) +no 𝐶) = ∩ {𝑥 ∈ On ∣ (∀𝑎 ∈ 𝐴 ((𝑎 +no 𝐵) +no 𝐶) ∈ 𝑥 ∧ ∀𝑏 ∈ 𝐵 ((𝐴 +no 𝑏) +no 𝐶) ∈ 𝑥 ∧ ∀𝑐 ∈ 𝐶 ((𝐴 +no 𝐵) +no 𝑐) ∈ 𝑥)})

Distinct variable groups:   𝐴,𝑎,𝑏,𝑐,𝑥   𝐵,𝑎,𝑏,𝑐,𝑥   𝐶,𝑎,𝑏,𝑐,𝑥

Proof of Theorem naddasslem1

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		naddcl 8644	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) ∈ On)
2	1	3adant3 1132	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 +no 𝐵) ∈ On)
3		simp3 1138	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐶 ∈ On)
4		naddov3 8647	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = ∩ {𝑎 ∈ On ∣ (( +no “ ({𝐴} × 𝐵)) ∪ ( +no “ (𝐴 × {𝐵}))) ⊆ 𝑎})
5	4	3adant3 1132	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 +no 𝐵) = ∩ {𝑎 ∈ On ∣ (( +no “ ({𝐴} × 𝐵)) ∪ ( +no “ (𝐴 × {𝐵}))) ⊆ 𝑎})
6		intmin 4935	. . . . 5 ⊢ (𝐶 ∈ On → ∩ {𝑐 ∈ On ∣ 𝐶 ⊆ 𝑐} = 𝐶)
7	6	eqcomd 2736	. . . 4 ⊢ (𝐶 ∈ On → 𝐶 = ∩ {𝑐 ∈ On ∣ 𝐶 ⊆ 𝑐})
8	7	3ad2ant3 1135	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐶 = ∩ {𝑐 ∈ On ∣ 𝐶 ⊆ 𝑐})
9	2, 3, 5, 8	naddunif 8660	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 +no 𝐵) +no 𝐶) = ∩ {𝑥 ∈ On ∣ (( +no “ ((( +no “ ({𝐴} × 𝐵)) ∪ ( +no “ (𝐴 × {𝐵}))) × {𝐶})) ∪ ( +no “ ({(𝐴 +no 𝐵)} × 𝐶))) ⊆ 𝑥})
10		df-3an 1088	. . . . . 6 ⊢ ((( +no “ (( +no “ (𝐴 × {𝐵})) × {𝐶})) ⊆ 𝑥 ∧ ( +no “ (( +no “ ({𝐴} × 𝐵)) × {𝐶})) ⊆ 𝑥 ∧ ( +no “ ({(𝐴 +no 𝐵)} × 𝐶)) ⊆ 𝑥) ↔ ((( +no “ (( +no “ (𝐴 × {𝐵})) × {𝐶})) ⊆ 𝑥 ∧ ( +no “ (( +no “ ({𝐴} × 𝐵)) × {𝐶})) ⊆ 𝑥) ∧ ( +no “ ({(𝐴 +no 𝐵)} × 𝐶)) ⊆ 𝑥))
11		unss 4156	. . . . . . . 8 ⊢ ((( +no “ (( +no “ ({𝐴} × 𝐵)) × {𝐶})) ⊆ 𝑥 ∧ ( +no “ (( +no “ (𝐴 × {𝐵})) × {𝐶})) ⊆ 𝑥) ↔ (( +no “ (( +no “ ({𝐴} × 𝐵)) × {𝐶})) ∪ ( +no “ (( +no “ (𝐴 × {𝐵})) × {𝐶}))) ⊆ 𝑥)
12		ancom 460	. . . . . . . 8 ⊢ ((( +no “ (( +no “ (𝐴 × {𝐵})) × {𝐶})) ⊆ 𝑥 ∧ ( +no “ (( +no “ ({𝐴} × 𝐵)) × {𝐶})) ⊆ 𝑥) ↔ (( +no “ (( +no “ ({𝐴} × 𝐵)) × {𝐶})) ⊆ 𝑥 ∧ ( +no “ (( +no “ (𝐴 × {𝐵})) × {𝐶})) ⊆ 𝑥))
13		xpundir 5711	. . . . . . . . . . 11 ⊢ ((( +no “ ({𝐴} × 𝐵)) ∪ ( +no “ (𝐴 × {𝐵}))) × {𝐶}) = ((( +no “ ({𝐴} × 𝐵)) × {𝐶}) ∪ (( +no “ (𝐴 × {𝐵})) × {𝐶}))
14	13	imaeq2i 6032	. . . . . . . . . 10 ⊢ ( +no “ ((( +no “ ({𝐴} × 𝐵)) ∪ ( +no “ (𝐴 × {𝐵}))) × {𝐶})) = ( +no “ ((( +no “ ({𝐴} × 𝐵)) × {𝐶}) ∪ (( +no “ (𝐴 × {𝐵})) × {𝐶})))
15		imaundi 6125	. . . . . . . . . 10 ⊢ ( +no “ ((( +no “ ({𝐴} × 𝐵)) × {𝐶}) ∪ (( +no “ (𝐴 × {𝐵})) × {𝐶}))) = (( +no “ (( +no “ ({𝐴} × 𝐵)) × {𝐶})) ∪ ( +no “ (( +no “ (𝐴 × {𝐵})) × {𝐶})))
16	14, 15	eqtri 2753	. . . . . . . . 9 ⊢ ( +no “ ((( +no “ ({𝐴} × 𝐵)) ∪ ( +no “ (𝐴 × {𝐵}))) × {𝐶})) = (( +no “ (( +no “ ({𝐴} × 𝐵)) × {𝐶})) ∪ ( +no “ (( +no “ (𝐴 × {𝐵})) × {𝐶})))
17	16	sseq1i 3978	. . . . . . . 8 ⊢ (( +no “ ((( +no “ ({𝐴} × 𝐵)) ∪ ( +no “ (𝐴 × {𝐵}))) × {𝐶})) ⊆ 𝑥 ↔ (( +no “ (( +no “ ({𝐴} × 𝐵)) × {𝐶})) ∪ ( +no “ (( +no “ (𝐴 × {𝐵})) × {𝐶}))) ⊆ 𝑥)
18	11, 12, 17	3bitr4i 303	. . . . . . 7 ⊢ ((( +no “ (( +no “ (𝐴 × {𝐵})) × {𝐶})) ⊆ 𝑥 ∧ ( +no “ (( +no “ ({𝐴} × 𝐵)) × {𝐶})) ⊆ 𝑥) ↔ ( +no “ ((( +no “ ({𝐴} × 𝐵)) ∪ ( +no “ (𝐴 × {𝐵}))) × {𝐶})) ⊆ 𝑥)
19	18	anbi1i 624	. . . . . 6 ⊢ (((( +no “ (( +no “ (𝐴 × {𝐵})) × {𝐶})) ⊆ 𝑥 ∧ ( +no “ (( +no “ ({𝐴} × 𝐵)) × {𝐶})) ⊆ 𝑥) ∧ ( +no “ ({(𝐴 +no 𝐵)} × 𝐶)) ⊆ 𝑥) ↔ (( +no “ ((( +no “ ({𝐴} × 𝐵)) ∪ ( +no “ (𝐴 × {𝐵}))) × {𝐶})) ⊆ 𝑥 ∧ ( +no “ ({(𝐴 +no 𝐵)} × 𝐶)) ⊆ 𝑥))
20		unss 4156	. . . . . 6 ⊢ ((( +no “ ((( +no “ ({𝐴} × 𝐵)) ∪ ( +no “ (𝐴 × {𝐵}))) × {𝐶})) ⊆ 𝑥 ∧ ( +no “ ({(𝐴 +no 𝐵)} × 𝐶)) ⊆ 𝑥) ↔ (( +no “ ((( +no “ ({𝐴} × 𝐵)) ∪ ( +no “ (𝐴 × {𝐵}))) × {𝐶})) ∪ ( +no “ ({(𝐴 +no 𝐵)} × 𝐶))) ⊆ 𝑥)
21	10, 19, 20	3bitrri 298	. . . . 5 ⊢ ((( +no “ ((( +no “ ({𝐴} × 𝐵)) ∪ ( +no “ (𝐴 × {𝐵}))) × {𝐶})) ∪ ( +no “ ({(𝐴 +no 𝐵)} × 𝐶))) ⊆ 𝑥 ↔ (( +no “ (( +no “ (𝐴 × {𝐵})) × {𝐶})) ⊆ 𝑥 ∧ ( +no “ (( +no “ ({𝐴} × 𝐵)) × {𝐶})) ⊆ 𝑥 ∧ ( +no “ ({(𝐴 +no 𝐵)} × 𝐶)) ⊆ 𝑥))
22		naddfn 8642	. . . . . . . . 9 ⊢ +no Fn (On × On)
23		fnfun 6621	. . . . . . . . 9 ⊢ ( +no Fn (On × On) → Fun +no )
24	22, 23	ax-mp 5	. . . . . . . 8 ⊢ Fun +no
25		imassrn 6045	. . . . . . . . . . 11 ⊢ ( +no “ (𝐴 × {𝐵})) ⊆ ran +no
26		naddf 8648	. . . . . . . . . . . 12 ⊢ +no :(On × On)⟶On
27		frn 6698	. . . . . . . . . . . 12 ⊢ ( +no :(On × On)⟶On → ran +no ⊆ On)
28	26, 27	ax-mp 5	. . . . . . . . . . 11 ⊢ ran +no ⊆ On
29	25, 28	sstri 3959	. . . . . . . . . 10 ⊢ ( +no “ (𝐴 × {𝐵})) ⊆ On
30		simpl3 1194	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → 𝐶 ∈ On)
31	30	snssd 4776	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → {𝐶} ⊆ On)
32		xpss12 5656	. . . . . . . . . 10 ⊢ ((( +no “ (𝐴 × {𝐵})) ⊆ On ∧ {𝐶} ⊆ On) → (( +no “ (𝐴 × {𝐵})) × {𝐶}) ⊆ (On × On))
33	29, 31, 32	sylancr 587	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (( +no “ (𝐴 × {𝐵})) × {𝐶}) ⊆ (On × On))
34	22	fndmi 6625	. . . . . . . . 9 ⊢ dom +no = (On × On)
35	33, 34	sseqtrrdi 3991	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (( +no “ (𝐴 × {𝐵})) × {𝐶}) ⊆ dom +no )
36		funimassov 7569	. . . . . . . 8 ⊢ ((Fun +no ∧ (( +no “ (𝐴 × {𝐵})) × {𝐶}) ⊆ dom +no ) → (( +no “ (( +no “ (𝐴 × {𝐵})) × {𝐶})) ⊆ 𝑥 ↔ ∀𝑝 ∈ ( +no “ (𝐴 × {𝐵}))∀𝑐 ∈ {𝐶} (𝑝 +no 𝑐) ∈ 𝑥))
37	24, 35, 36	sylancr 587	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (( +no “ (( +no “ (𝐴 × {𝐵})) × {𝐶})) ⊆ 𝑥 ↔ ∀𝑝 ∈ ( +no “ (𝐴 × {𝐵}))∀𝑐 ∈ {𝐶} (𝑝 +no 𝑐) ∈ 𝑥))
38		oveq2 7398	. . . . . . . . . . 11 ⊢ (𝑐 = 𝐶 → (𝑝 +no 𝑐) = (𝑝 +no 𝐶))
39	38	eleq1d 2814	. . . . . . . . . 10 ⊢ (𝑐 = 𝐶 → ((𝑝 +no 𝑐) ∈ 𝑥 ↔ (𝑝 +no 𝐶) ∈ 𝑥))
40	39	ralsng 4642	. . . . . . . . 9 ⊢ (𝐶 ∈ On → (∀𝑐 ∈ {𝐶} (𝑝 +no 𝑐) ∈ 𝑥 ↔ (𝑝 +no 𝐶) ∈ 𝑥))
41	30, 40	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (∀𝑐 ∈ {𝐶} (𝑝 +no 𝑐) ∈ 𝑥 ↔ (𝑝 +no 𝐶) ∈ 𝑥))
42	41	ralbidv 3157	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (∀𝑝 ∈ ( +no “ (𝐴 × {𝐵}))∀𝑐 ∈ {𝐶} (𝑝 +no 𝑐) ∈ 𝑥 ↔ ∀𝑝 ∈ ( +no “ (𝐴 × {𝐵}))(𝑝 +no 𝐶) ∈ 𝑥))
43		onss 7764	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
44	43	3ad2ant1 1133	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐴 ⊆ On)
45	44	adantr 480	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → 𝐴 ⊆ On)
46		simpl2 1193	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → 𝐵 ∈ On)
47	46	snssd 4776	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → {𝐵} ⊆ On)
48		xpss12 5656	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ On ∧ {𝐵} ⊆ On) → (𝐴 × {𝐵}) ⊆ (On × On))
49	45, 47, 48	syl2anc 584	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (𝐴 × {𝐵}) ⊆ (On × On))
50		oveq1 7397	. . . . . . . . . . 11 ⊢ (𝑝 = (𝑎 +no 𝑏) → (𝑝 +no 𝐶) = ((𝑎 +no 𝑏) +no 𝐶))
51	50	eleq1d 2814	. . . . . . . . . 10 ⊢ (𝑝 = (𝑎 +no 𝑏) → ((𝑝 +no 𝐶) ∈ 𝑥 ↔ ((𝑎 +no 𝑏) +no 𝐶) ∈ 𝑥))
52	51	imaeqalov 7631	. . . . . . . . 9 ⊢ (( +no Fn (On × On) ∧ (𝐴 × {𝐵}) ⊆ (On × On)) → (∀𝑝 ∈ ( +no “ (𝐴 × {𝐵}))(𝑝 +no 𝐶) ∈ 𝑥 ↔ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ {𝐵} ((𝑎 +no 𝑏) +no 𝐶) ∈ 𝑥))
53	22, 49, 52	sylancr 587	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (∀𝑝 ∈ ( +no “ (𝐴 × {𝐵}))(𝑝 +no 𝐶) ∈ 𝑥 ↔ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ {𝐵} ((𝑎 +no 𝑏) +no 𝐶) ∈ 𝑥))
54		oveq2 7398	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝐵 → (𝑎 +no 𝑏) = (𝑎 +no 𝐵))
55	54	oveq1d 7405	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝐵 → ((𝑎 +no 𝑏) +no 𝐶) = ((𝑎 +no 𝐵) +no 𝐶))
56	55	eleq1d 2814	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐵 → (((𝑎 +no 𝑏) +no 𝐶) ∈ 𝑥 ↔ ((𝑎 +no 𝐵) +no 𝐶) ∈ 𝑥))
57	56	ralsng 4642	. . . . . . . . . 10 ⊢ (𝐵 ∈ On → (∀𝑏 ∈ {𝐵} ((𝑎 +no 𝑏) +no 𝐶) ∈ 𝑥 ↔ ((𝑎 +no 𝐵) +no 𝐶) ∈ 𝑥))
58	46, 57	syl 17	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (∀𝑏 ∈ {𝐵} ((𝑎 +no 𝑏) +no 𝐶) ∈ 𝑥 ↔ ((𝑎 +no 𝐵) +no 𝐶) ∈ 𝑥))
59	58	ralbidv 3157	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (∀𝑎 ∈ 𝐴 ∀𝑏 ∈ {𝐵} ((𝑎 +no 𝑏) +no 𝐶) ∈ 𝑥 ↔ ∀𝑎 ∈ 𝐴 ((𝑎 +no 𝐵) +no 𝐶) ∈ 𝑥))
60	53, 59	bitrd 279	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (∀𝑝 ∈ ( +no “ (𝐴 × {𝐵}))(𝑝 +no 𝐶) ∈ 𝑥 ↔ ∀𝑎 ∈ 𝐴 ((𝑎 +no 𝐵) +no 𝐶) ∈ 𝑥))
61	37, 42, 60	3bitrd 305	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (( +no “ (( +no “ (𝐴 × {𝐵})) × {𝐶})) ⊆ 𝑥 ↔ ∀𝑎 ∈ 𝐴 ((𝑎 +no 𝐵) +no 𝐶) ∈ 𝑥))
62		imassrn 6045	. . . . . . . . . . 11 ⊢ ( +no “ ({𝐴} × 𝐵)) ⊆ ran +no
63	62, 28	sstri 3959	. . . . . . . . . 10 ⊢ ( +no “ ({𝐴} × 𝐵)) ⊆ On
64		xpss12 5656	. . . . . . . . . 10 ⊢ ((( +no “ ({𝐴} × 𝐵)) ⊆ On ∧ {𝐶} ⊆ On) → (( +no “ ({𝐴} × 𝐵)) × {𝐶}) ⊆ (On × On))
65	63, 31, 64	sylancr 587	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (( +no “ ({𝐴} × 𝐵)) × {𝐶}) ⊆ (On × On))
66	65, 34	sseqtrrdi 3991	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (( +no “ ({𝐴} × 𝐵)) × {𝐶}) ⊆ dom +no )
67		funimassov 7569	. . . . . . . 8 ⊢ ((Fun +no ∧ (( +no “ ({𝐴} × 𝐵)) × {𝐶}) ⊆ dom +no ) → (( +no “ (( +no “ ({𝐴} × 𝐵)) × {𝐶})) ⊆ 𝑥 ↔ ∀𝑝 ∈ ( +no “ ({𝐴} × 𝐵))∀𝑐 ∈ {𝐶} (𝑝 +no 𝑐) ∈ 𝑥))
68	24, 66, 67	sylancr 587	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (( +no “ (( +no “ ({𝐴} × 𝐵)) × {𝐶})) ⊆ 𝑥 ↔ ∀𝑝 ∈ ( +no “ ({𝐴} × 𝐵))∀𝑐 ∈ {𝐶} (𝑝 +no 𝑐) ∈ 𝑥))
69	41	ralbidv 3157	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (∀𝑝 ∈ ( +no “ ({𝐴} × 𝐵))∀𝑐 ∈ {𝐶} (𝑝 +no 𝑐) ∈ 𝑥 ↔ ∀𝑝 ∈ ( +no “ ({𝐴} × 𝐵))(𝑝 +no 𝐶) ∈ 𝑥))
70		simpl1 1192	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → 𝐴 ∈ On)
71	70	snssd 4776	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → {𝐴} ⊆ On)
72		onss 7764	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On)
73	72	3ad2ant2 1134	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐵 ⊆ On)
74	73	adantr 480	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → 𝐵 ⊆ On)
75		xpss12 5656	. . . . . . . . . 10 ⊢ (({𝐴} ⊆ On ∧ 𝐵 ⊆ On) → ({𝐴} × 𝐵) ⊆ (On × On))
76	71, 74, 75	syl2anc 584	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → ({𝐴} × 𝐵) ⊆ (On × On))
77	51	imaeqalov 7631	. . . . . . . . 9 ⊢ (( +no Fn (On × On) ∧ ({𝐴} × 𝐵) ⊆ (On × On)) → (∀𝑝 ∈ ( +no “ ({𝐴} × 𝐵))(𝑝 +no 𝐶) ∈ 𝑥 ↔ ∀𝑎 ∈ {𝐴}∀𝑏 ∈ 𝐵 ((𝑎 +no 𝑏) +no 𝐶) ∈ 𝑥))
78	22, 76, 77	sylancr 587	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (∀𝑝 ∈ ( +no “ ({𝐴} × 𝐵))(𝑝 +no 𝐶) ∈ 𝑥 ↔ ∀𝑎 ∈ {𝐴}∀𝑏 ∈ 𝐵 ((𝑎 +no 𝑏) +no 𝐶) ∈ 𝑥))
79		oveq1 7397	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝐴 → (𝑎 +no 𝑏) = (𝐴 +no 𝑏))
80	79	oveq1d 7405	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → ((𝑎 +no 𝑏) +no 𝐶) = ((𝐴 +no 𝑏) +no 𝐶))
81	80	eleq1d 2814	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (((𝑎 +no 𝑏) +no 𝐶) ∈ 𝑥 ↔ ((𝐴 +no 𝑏) +no 𝐶) ∈ 𝑥))
82	81	ralbidv 3157	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (∀𝑏 ∈ 𝐵 ((𝑎 +no 𝑏) +no 𝐶) ∈ 𝑥 ↔ ∀𝑏 ∈ 𝐵 ((𝐴 +no 𝑏) +no 𝐶) ∈ 𝑥))
83	82	ralsng 4642	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (∀𝑎 ∈ {𝐴}∀𝑏 ∈ 𝐵 ((𝑎 +no 𝑏) +no 𝐶) ∈ 𝑥 ↔ ∀𝑏 ∈ 𝐵 ((𝐴 +no 𝑏) +no 𝐶) ∈ 𝑥))
84	70, 83	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (∀𝑎 ∈ {𝐴}∀𝑏 ∈ 𝐵 ((𝑎 +no 𝑏) +no 𝐶) ∈ 𝑥 ↔ ∀𝑏 ∈ 𝐵 ((𝐴 +no 𝑏) +no 𝐶) ∈ 𝑥))
85	78, 84	bitrd 279	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (∀𝑝 ∈ ( +no “ ({𝐴} × 𝐵))(𝑝 +no 𝐶) ∈ 𝑥 ↔ ∀𝑏 ∈ 𝐵 ((𝐴 +no 𝑏) +no 𝐶) ∈ 𝑥))
86	68, 69, 85	3bitrd 305	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (( +no “ (( +no “ ({𝐴} × 𝐵)) × {𝐶})) ⊆ 𝑥 ↔ ∀𝑏 ∈ 𝐵 ((𝐴 +no 𝑏) +no 𝐶) ∈ 𝑥))
87	2	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (𝐴 +no 𝐵) ∈ On)
88	87	snssd 4776	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → {(𝐴 +no 𝐵)} ⊆ On)
89		onss 7764	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ On → 𝐶 ⊆ On)
90	89	3ad2ant3 1135	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐶 ⊆ On)
91	90	adantr 480	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → 𝐶 ⊆ On)
92		xpss12 5656	. . . . . . . . . 10 ⊢ (({(𝐴 +no 𝐵)} ⊆ On ∧ 𝐶 ⊆ On) → ({(𝐴 +no 𝐵)} × 𝐶) ⊆ (On × On))
93	88, 91, 92	syl2anc 584	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → ({(𝐴 +no 𝐵)} × 𝐶) ⊆ (On × On))
94	93, 34	sseqtrrdi 3991	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → ({(𝐴 +no 𝐵)} × 𝐶) ⊆ dom +no )
95		funimassov 7569	. . . . . . . 8 ⊢ ((Fun +no ∧ ({(𝐴 +no 𝐵)} × 𝐶) ⊆ dom +no ) → (( +no “ ({(𝐴 +no 𝐵)} × 𝐶)) ⊆ 𝑥 ↔ ∀𝑎 ∈ {(𝐴 +no 𝐵)}∀𝑐 ∈ 𝐶 (𝑎 +no 𝑐) ∈ 𝑥))
96	24, 94, 95	sylancr 587	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (( +no “ ({(𝐴 +no 𝐵)} × 𝐶)) ⊆ 𝑥 ↔ ∀𝑎 ∈ {(𝐴 +no 𝐵)}∀𝑐 ∈ 𝐶 (𝑎 +no 𝑐) ∈ 𝑥))
97		ovex 7423	. . . . . . . 8 ⊢ (𝐴 +no 𝐵) ∈ V
98		oveq1 7397	. . . . . . . . . 10 ⊢ (𝑎 = (𝐴 +no 𝐵) → (𝑎 +no 𝑐) = ((𝐴 +no 𝐵) +no 𝑐))
99	98	eleq1d 2814	. . . . . . . . 9 ⊢ (𝑎 = (𝐴 +no 𝐵) → ((𝑎 +no 𝑐) ∈ 𝑥 ↔ ((𝐴 +no 𝐵) +no 𝑐) ∈ 𝑥))
100	99	ralbidv 3157	. . . . . . . 8 ⊢ (𝑎 = (𝐴 +no 𝐵) → (∀𝑐 ∈ 𝐶 (𝑎 +no 𝑐) ∈ 𝑥 ↔ ∀𝑐 ∈ 𝐶 ((𝐴 +no 𝐵) +no 𝑐) ∈ 𝑥))
101	97, 100	ralsn 4648	. . . . . . 7 ⊢ (∀𝑎 ∈ {(𝐴 +no 𝐵)}∀𝑐 ∈ 𝐶 (𝑎 +no 𝑐) ∈ 𝑥 ↔ ∀𝑐 ∈ 𝐶 ((𝐴 +no 𝐵) +no 𝑐) ∈ 𝑥)
102	96, 101	bitrdi 287	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (( +no “ ({(𝐴 +no 𝐵)} × 𝐶)) ⊆ 𝑥 ↔ ∀𝑐 ∈ 𝐶 ((𝐴 +no 𝐵) +no 𝑐) ∈ 𝑥))
103	61, 86, 102	3anbi123d 1438	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → ((( +no “ (( +no “ (𝐴 × {𝐵})) × {𝐶})) ⊆ 𝑥 ∧ ( +no “ (( +no “ ({𝐴} × 𝐵)) × {𝐶})) ⊆ 𝑥 ∧ ( +no “ ({(𝐴 +no 𝐵)} × 𝐶)) ⊆ 𝑥) ↔ (∀𝑎 ∈ 𝐴 ((𝑎 +no 𝐵) +no 𝐶) ∈ 𝑥 ∧ ∀𝑏 ∈ 𝐵 ((𝐴 +no 𝑏) +no 𝐶) ∈ 𝑥 ∧ ∀𝑐 ∈ 𝐶 ((𝐴 +no 𝐵) +no 𝑐) ∈ 𝑥)))
104	21, 103	bitrid 283	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → ((( +no “ ((( +no “ ({𝐴} × 𝐵)) ∪ ( +no “ (𝐴 × {𝐵}))) × {𝐶})) ∪ ( +no “ ({(𝐴 +no 𝐵)} × 𝐶))) ⊆ 𝑥 ↔ (∀𝑎 ∈ 𝐴 ((𝑎 +no 𝐵) +no 𝐶) ∈ 𝑥 ∧ ∀𝑏 ∈ 𝐵 ((𝐴 +no 𝑏) +no 𝐶) ∈ 𝑥 ∧ ∀𝑐 ∈ 𝐶 ((𝐴 +no 𝐵) +no 𝑐) ∈ 𝑥)))
105	104	rabbidva 3415	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → {𝑥 ∈ On ∣ (( +no “ ((( +no “ ({𝐴} × 𝐵)) ∪ ( +no “ (𝐴 × {𝐵}))) × {𝐶})) ∪ ( +no “ ({(𝐴 +no 𝐵)} × 𝐶))) ⊆ 𝑥} = {𝑥 ∈ On ∣ (∀𝑎 ∈ 𝐴 ((𝑎 +no 𝐵) +no 𝐶) ∈ 𝑥 ∧ ∀𝑏 ∈ 𝐵 ((𝐴 +no 𝑏) +no 𝐶) ∈ 𝑥 ∧ ∀𝑐 ∈ 𝐶 ((𝐴 +no 𝐵) +no 𝑐) ∈ 𝑥)})
106	105	inteqd 4918	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ∩ {𝑥 ∈ On ∣ (( +no “ ((( +no “ ({𝐴} × 𝐵)) ∪ ( +no “ (𝐴 × {𝐵}))) × {𝐶})) ∪ ( +no “ ({(𝐴 +no 𝐵)} × 𝐶))) ⊆ 𝑥} = ∩ {𝑥 ∈ On ∣ (∀𝑎 ∈ 𝐴 ((𝑎 +no 𝐵) +no 𝐶) ∈ 𝑥 ∧ ∀𝑏 ∈ 𝐵 ((𝐴 +no 𝑏) +no 𝐶) ∈ 𝑥 ∧ ∀𝑐 ∈ 𝐶 ((𝐴 +no 𝐵) +no 𝑐) ∈ 𝑥)})
107	9, 106	eqtrd 2765	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 +no 𝐵) +no 𝐶) = ∩ {𝑥 ∈ On ∣ (∀𝑎 ∈ 𝐴 ((𝑎 +no 𝐵) +no 𝐶) ∈ 𝑥 ∧ ∀𝑏 ∈ 𝐵 ((𝐴 +no 𝑏) +no 𝐶) ∈ 𝑥 ∧ ∀𝑐 ∈ 𝐶 ((𝐴 +no 𝐵) +no 𝑐) ∈ 𝑥)})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3045  {crab 3408   ∪ cun 3915   ⊆ wss 3917  {csn 4592  ∩ cint 4913   × cxp 5639  dom cdm 5641  ran crn 5642   “ cima 5644  Oncon0 6335  Fun wfun 6508   Fn wfn 6509  ⟶wf 6510  (class class class)co 7390   +no cnadd 8632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-frecs 8263  df-nadd 8633

This theorem is referenced by:  naddass  8663