Theorem naddasslem2 8733

Description: Lemma for naddass 8734. Expand out the expression for natural addition of three arguments. (Contributed by Scott Fenton, 20-Jan-2025.)

Assertion

Ref	Expression
naddasslem2	⊢ ((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 naddasslem2

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1137	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐴 ∈ On)
2		naddcl 8715	. . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 +no 𝐶) ∈ On)
3	2	3adant1 1131	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 +no 𝐶) ∈ On)
4		intmin 4968	. . . . 5 ⊢ (𝐴 ∈ On → ∩ {𝑎 ∈ On ∣ 𝐴 ⊆ 𝑎} = 𝐴)
5	4	eqcomd 2743	. . . 4 ⊢ (𝐴 ∈ On → 𝐴 = ∩ {𝑎 ∈ On ∣ 𝐴 ⊆ 𝑎})
6	5	3ad2ant1 1134	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐴 = ∩ {𝑎 ∈ On ∣ 𝐴 ⊆ 𝑎})
7		naddov3 8718	. . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 +no 𝐶) = ∩ {𝑝 ∈ On ∣ (( +no “ ({𝐵} × 𝐶)) ∪ ( +no “ (𝐵 × {𝐶}))) ⊆ 𝑝})
8	7	3adant1 1131	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 +no 𝐶) = ∩ {𝑝 ∈ On ∣ (( +no “ ({𝐵} × 𝐶)) ∪ ( +no “ (𝐵 × {𝐶}))) ⊆ 𝑝})
9	1, 3, 6, 8	naddunif 8731	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 +no (𝐵 +no 𝐶)) = ∩ {𝑥 ∈ On ∣ (( +no “ (𝐴 × {(𝐵 +no 𝐶)})) ∪ ( +no “ ({𝐴} × (( +no “ ({𝐵} × 𝐶)) ∪ ( +no “ (𝐵 × {𝐶})))))) ⊆ 𝑥})
10		3anass 1095	. . . . . 6 ⊢ ((( +no “ (𝐴 × {(𝐵 +no 𝐶)})) ⊆ 𝑥 ∧ ( +no “ ({𝐴} × ( +no “ (𝐵 × {𝐶})))) ⊆ 𝑥 ∧ ( +no “ ({𝐴} × ( +no “ ({𝐵} × 𝐶)))) ⊆ 𝑥) ↔ (( +no “ (𝐴 × {(𝐵 +no 𝐶)})) ⊆ 𝑥 ∧ (( +no “ ({𝐴} × ( +no “ (𝐵 × {𝐶})))) ⊆ 𝑥 ∧ ( +no “ ({𝐴} × ( +no “ ({𝐵} × 𝐶)))) ⊆ 𝑥)))
11		unss 4190	. . . . . . . 8 ⊢ ((( +no “ ({𝐴} × ( +no “ ({𝐵} × 𝐶)))) ⊆ 𝑥 ∧ ( +no “ ({𝐴} × ( +no “ (𝐵 × {𝐶})))) ⊆ 𝑥) ↔ (( +no “ ({𝐴} × ( +no “ ({𝐵} × 𝐶)))) ∪ ( +no “ ({𝐴} × ( +no “ (𝐵 × {𝐶}))))) ⊆ 𝑥)
12		ancom 460	. . . . . . . 8 ⊢ ((( +no “ ({𝐴} × ( +no “ (𝐵 × {𝐶})))) ⊆ 𝑥 ∧ ( +no “ ({𝐴} × ( +no “ ({𝐵} × 𝐶)))) ⊆ 𝑥) ↔ (( +no “ ({𝐴} × ( +no “ ({𝐵} × 𝐶)))) ⊆ 𝑥 ∧ ( +no “ ({𝐴} × ( +no “ (𝐵 × {𝐶})))) ⊆ 𝑥))
13		xpundi 5754	. . . . . . . . . . 11 ⊢ ({𝐴} × (( +no “ ({𝐵} × 𝐶)) ∪ ( +no “ (𝐵 × {𝐶})))) = (({𝐴} × ( +no “ ({𝐵} × 𝐶))) ∪ ({𝐴} × ( +no “ (𝐵 × {𝐶}))))
14	13	imaeq2i 6076	. . . . . . . . . 10 ⊢ ( +no “ ({𝐴} × (( +no “ ({𝐵} × 𝐶)) ∪ ( +no “ (𝐵 × {𝐶}))))) = ( +no “ (({𝐴} × ( +no “ ({𝐵} × 𝐶))) ∪ ({𝐴} × ( +no “ (𝐵 × {𝐶})))))
15		imaundi 6169	. . . . . . . . . 10 ⊢ ( +no “ (({𝐴} × ( +no “ ({𝐵} × 𝐶))) ∪ ({𝐴} × ( +no “ (𝐵 × {𝐶}))))) = (( +no “ ({𝐴} × ( +no “ ({𝐵} × 𝐶)))) ∪ ( +no “ ({𝐴} × ( +no “ (𝐵 × {𝐶})))))
16	14, 15	eqtri 2765	. . . . . . . . 9 ⊢ ( +no “ ({𝐴} × (( +no “ ({𝐵} × 𝐶)) ∪ ( +no “ (𝐵 × {𝐶}))))) = (( +no “ ({𝐴} × ( +no “ ({𝐵} × 𝐶)))) ∪ ( +no “ ({𝐴} × ( +no “ (𝐵 × {𝐶})))))
17	16	sseq1i 4012	. . . . . . . 8 ⊢ (( +no “ ({𝐴} × (( +no “ ({𝐵} × 𝐶)) ∪ ( +no “ (𝐵 × {𝐶}))))) ⊆ 𝑥 ↔ (( +no “ ({𝐴} × ( +no “ ({𝐵} × 𝐶)))) ∪ ( +no “ ({𝐴} × ( +no “ (𝐵 × {𝐶}))))) ⊆ 𝑥)
18	11, 12, 17	3bitr4i 303	. . . . . . 7 ⊢ ((( +no “ ({𝐴} × ( +no “ (𝐵 × {𝐶})))) ⊆ 𝑥 ∧ ( +no “ ({𝐴} × ( +no “ ({𝐵} × 𝐶)))) ⊆ 𝑥) ↔ ( +no “ ({𝐴} × (( +no “ ({𝐵} × 𝐶)) ∪ ( +no “ (𝐵 × {𝐶}))))) ⊆ 𝑥)
19	18	anbi2i 623	. . . . . 6 ⊢ ((( +no “ (𝐴 × {(𝐵 +no 𝐶)})) ⊆ 𝑥 ∧ (( +no “ ({𝐴} × ( +no “ (𝐵 × {𝐶})))) ⊆ 𝑥 ∧ ( +no “ ({𝐴} × ( +no “ ({𝐵} × 𝐶)))) ⊆ 𝑥)) ↔ (( +no “ (𝐴 × {(𝐵 +no 𝐶)})) ⊆ 𝑥 ∧ ( +no “ ({𝐴} × (( +no “ ({𝐵} × 𝐶)) ∪ ( +no “ (𝐵 × {𝐶}))))) ⊆ 𝑥))
20		unss 4190	. . . . . 6 ⊢ ((( +no “ (𝐴 × {(𝐵 +no 𝐶)})) ⊆ 𝑥 ∧ ( +no “ ({𝐴} × (( +no “ ({𝐵} × 𝐶)) ∪ ( +no “ (𝐵 × {𝐶}))))) ⊆ 𝑥) ↔ (( +no “ (𝐴 × {(𝐵 +no 𝐶)})) ∪ ( +no “ ({𝐴} × (( +no “ ({𝐵} × 𝐶)) ∪ ( +no “ (𝐵 × {𝐶})))))) ⊆ 𝑥)
21	10, 19, 20	3bitrri 298	. . . . 5 ⊢ ((( +no “ (𝐴 × {(𝐵 +no 𝐶)})) ∪ ( +no “ ({𝐴} × (( +no “ ({𝐵} × 𝐶)) ∪ ( +no “ (𝐵 × {𝐶})))))) ⊆ 𝑥 ↔ (( +no “ (𝐴 × {(𝐵 +no 𝐶)})) ⊆ 𝑥 ∧ ( +no “ ({𝐴} × ( +no “ (𝐵 × {𝐶})))) ⊆ 𝑥 ∧ ( +no “ ({𝐴} × ( +no “ ({𝐵} × 𝐶)))) ⊆ 𝑥))
22		naddfn 8713	. . . . . . . . 9 ⊢ +no Fn (On × On)
23		fnfun 6668	. . . . . . . . 9 ⊢ ( +no Fn (On × On) → Fun +no )
24	22, 23	ax-mp 5	. . . . . . . 8 ⊢ Fun +no
25		onss 7805	. . . . . . . . . . 11 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
26	25	3ad2ant1 1134	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐴 ⊆ On)
27	3	adantr 480	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (𝐵 +no 𝐶) ∈ On)
28	27	snssd 4809	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → {(𝐵 +no 𝐶)} ⊆ On)
29		xpss12 5700	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ On ∧ {(𝐵 +no 𝐶)} ⊆ On) → (𝐴 × {(𝐵 +no 𝐶)}) ⊆ (On × On))
30	26, 28, 29	syl2an2r 685	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (𝐴 × {(𝐵 +no 𝐶)}) ⊆ (On × On))
31	22	fndmi 6672	. . . . . . . . 9 ⊢ dom +no = (On × On)
32	30, 31	sseqtrrdi 4025	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (𝐴 × {(𝐵 +no 𝐶)}) ⊆ dom +no )
33		funimassov 7610	. . . . . . . 8 ⊢ ((Fun +no ∧ (𝐴 × {(𝐵 +no 𝐶)}) ⊆ dom +no ) → (( +no “ (𝐴 × {(𝐵 +no 𝐶)})) ⊆ 𝑥 ↔ ∀𝑎 ∈ 𝐴 ∀𝑝 ∈ {(𝐵 +no 𝐶)} (𝑎 +no 𝑝) ∈ 𝑥))
34	24, 32, 33	sylancr 587	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (( +no “ (𝐴 × {(𝐵 +no 𝐶)})) ⊆ 𝑥 ↔ ∀𝑎 ∈ 𝐴 ∀𝑝 ∈ {(𝐵 +no 𝐶)} (𝑎 +no 𝑝) ∈ 𝑥))
35		ovex 7464	. . . . . . . . 9 ⊢ (𝐵 +no 𝐶) ∈ V
36		oveq2 7439	. . . . . . . . . 10 ⊢ (𝑝 = (𝐵 +no 𝐶) → (𝑎 +no 𝑝) = (𝑎 +no (𝐵 +no 𝐶)))
37	36	eleq1d 2826	. . . . . . . . 9 ⊢ (𝑝 = (𝐵 +no 𝐶) → ((𝑎 +no 𝑝) ∈ 𝑥 ↔ (𝑎 +no (𝐵 +no 𝐶)) ∈ 𝑥))
38	35, 37	ralsn 4681	. . . . . . . 8 ⊢ (∀𝑝 ∈ {(𝐵 +no 𝐶)} (𝑎 +no 𝑝) ∈ 𝑥 ↔ (𝑎 +no (𝐵 +no 𝐶)) ∈ 𝑥)
39	38	ralbii 3093	. . . . . . 7 ⊢ (∀𝑎 ∈ 𝐴 ∀𝑝 ∈ {(𝐵 +no 𝐶)} (𝑎 +no 𝑝) ∈ 𝑥 ↔ ∀𝑎 ∈ 𝐴 (𝑎 +no (𝐵 +no 𝐶)) ∈ 𝑥)
40	34, 39	bitrdi 287	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (( +no “ (𝐴 × {(𝐵 +no 𝐶)})) ⊆ 𝑥 ↔ ∀𝑎 ∈ 𝐴 (𝑎 +no (𝐵 +no 𝐶)) ∈ 𝑥))
41		simpl1 1192	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → 𝐴 ∈ On)
42	41	snssd 4809	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → {𝐴} ⊆ On)
43		imassrn 6089	. . . . . . . . . . 11 ⊢ ( +no “ (𝐵 × {𝐶})) ⊆ ran +no
44		naddf 8719	. . . . . . . . . . . 12 ⊢ +no :(On × On)⟶On
45		frn 6743	. . . . . . . . . . . 12 ⊢ ( +no :(On × On)⟶On → ran +no ⊆ On)
46	44, 45	ax-mp 5	. . . . . . . . . . 11 ⊢ ran +no ⊆ On
47	43, 46	sstri 3993	. . . . . . . . . 10 ⊢ ( +no “ (𝐵 × {𝐶})) ⊆ On
48		xpss12 5700	. . . . . . . . . 10 ⊢ (({𝐴} ⊆ On ∧ ( +no “ (𝐵 × {𝐶})) ⊆ On) → ({𝐴} × ( +no “ (𝐵 × {𝐶}))) ⊆ (On × On))
49	42, 47, 48	sylancl 586	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → ({𝐴} × ( +no “ (𝐵 × {𝐶}))) ⊆ (On × On))
50	49, 31	sseqtrrdi 4025	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → ({𝐴} × ( +no “ (𝐵 × {𝐶}))) ⊆ dom +no )
51		funimassov 7610	. . . . . . . 8 ⊢ ((Fun +no ∧ ({𝐴} × ( +no “ (𝐵 × {𝐶}))) ⊆ dom +no ) → (( +no “ ({𝐴} × ( +no “ (𝐵 × {𝐶})))) ⊆ 𝑥 ↔ ∀𝑎 ∈ {𝐴}∀𝑝 ∈ ( +no “ (𝐵 × {𝐶}))(𝑎 +no 𝑝) ∈ 𝑥))
52	24, 50, 51	sylancr 587	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (( +no “ ({𝐴} × ( +no “ (𝐵 × {𝐶})))) ⊆ 𝑥 ↔ ∀𝑎 ∈ {𝐴}∀𝑝 ∈ ( +no “ (𝐵 × {𝐶}))(𝑎 +no 𝑝) ∈ 𝑥))
53		oveq1 7438	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (𝑎 +no 𝑝) = (𝐴 +no 𝑝))
54	53	eleq1d 2826	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → ((𝑎 +no 𝑝) ∈ 𝑥 ↔ (𝐴 +no 𝑝) ∈ 𝑥))
55	54	ralbidv 3178	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (∀𝑝 ∈ ( +no “ (𝐵 × {𝐶}))(𝑎 +no 𝑝) ∈ 𝑥 ↔ ∀𝑝 ∈ ( +no “ (𝐵 × {𝐶}))(𝐴 +no 𝑝) ∈ 𝑥))
56	55	ralsng 4675	. . . . . . . 8 ⊢ (𝐴 ∈ On → (∀𝑎 ∈ {𝐴}∀𝑝 ∈ ( +no “ (𝐵 × {𝐶}))(𝑎 +no 𝑝) ∈ 𝑥 ↔ ∀𝑝 ∈ ( +no “ (𝐵 × {𝐶}))(𝐴 +no 𝑝) ∈ 𝑥))
57	41, 56	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (∀𝑎 ∈ {𝐴}∀𝑝 ∈ ( +no “ (𝐵 × {𝐶}))(𝑎 +no 𝑝) ∈ 𝑥 ↔ ∀𝑝 ∈ ( +no “ (𝐵 × {𝐶}))(𝐴 +no 𝑝) ∈ 𝑥))
58		onss 7805	. . . . . . . . . . 11 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On)
59	58	3ad2ant2 1135	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐵 ⊆ On)
60		simpl3 1194	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → 𝐶 ∈ On)
61	60	snssd 4809	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → {𝐶} ⊆ On)
62		xpss12 5700	. . . . . . . . . 10 ⊢ ((𝐵 ⊆ On ∧ {𝐶} ⊆ On) → (𝐵 × {𝐶}) ⊆ (On × On))
63	59, 61, 62	syl2an2r 685	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (𝐵 × {𝐶}) ⊆ (On × On))
64		oveq2 7439	. . . . . . . . . . 11 ⊢ (𝑝 = (𝑏 +no 𝑐) → (𝐴 +no 𝑝) = (𝐴 +no (𝑏 +no 𝑐)))
65	64	eleq1d 2826	. . . . . . . . . 10 ⊢ (𝑝 = (𝑏 +no 𝑐) → ((𝐴 +no 𝑝) ∈ 𝑥 ↔ (𝐴 +no (𝑏 +no 𝑐)) ∈ 𝑥))
66	65	imaeqalov 7672	. . . . . . . . 9 ⊢ (( +no Fn (On × On) ∧ (𝐵 × {𝐶}) ⊆ (On × On)) → (∀𝑝 ∈ ( +no “ (𝐵 × {𝐶}))(𝐴 +no 𝑝) ∈ 𝑥 ↔ ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ {𝐶} (𝐴 +no (𝑏 +no 𝑐)) ∈ 𝑥))
67	22, 63, 66	sylancr 587	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (∀𝑝 ∈ ( +no “ (𝐵 × {𝐶}))(𝐴 +no 𝑝) ∈ 𝑥 ↔ ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ {𝐶} (𝐴 +no (𝑏 +no 𝑐)) ∈ 𝑥))
68		oveq2 7439	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝐶 → (𝑏 +no 𝑐) = (𝑏 +no 𝐶))
69	68	oveq2d 7447	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝐶 → (𝐴 +no (𝑏 +no 𝑐)) = (𝐴 +no (𝑏 +no 𝐶)))
70	69	eleq1d 2826	. . . . . . . . . . 11 ⊢ (𝑐 = 𝐶 → ((𝐴 +no (𝑏 +no 𝑐)) ∈ 𝑥 ↔ (𝐴 +no (𝑏 +no 𝐶)) ∈ 𝑥))
71	70	ralsng 4675	. . . . . . . . . 10 ⊢ (𝐶 ∈ On → (∀𝑐 ∈ {𝐶} (𝐴 +no (𝑏 +no 𝑐)) ∈ 𝑥 ↔ (𝐴 +no (𝑏 +no 𝐶)) ∈ 𝑥))
72	60, 71	syl 17	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (∀𝑐 ∈ {𝐶} (𝐴 +no (𝑏 +no 𝑐)) ∈ 𝑥 ↔ (𝐴 +no (𝑏 +no 𝐶)) ∈ 𝑥))
73	72	ralbidv 3178	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (∀𝑏 ∈ 𝐵 ∀𝑐 ∈ {𝐶} (𝐴 +no (𝑏 +no 𝑐)) ∈ 𝑥 ↔ ∀𝑏 ∈ 𝐵 (𝐴 +no (𝑏 +no 𝐶)) ∈ 𝑥))
74	67, 73	bitrd 279	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (∀𝑝 ∈ ( +no “ (𝐵 × {𝐶}))(𝐴 +no 𝑝) ∈ 𝑥 ↔ ∀𝑏 ∈ 𝐵 (𝐴 +no (𝑏 +no 𝐶)) ∈ 𝑥))
75	52, 57, 74	3bitrd 305	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (( +no “ ({𝐴} × ( +no “ (𝐵 × {𝐶})))) ⊆ 𝑥 ↔ ∀𝑏 ∈ 𝐵 (𝐴 +no (𝑏 +no 𝐶)) ∈ 𝑥))
76		imassrn 6089	. . . . . . . . . . 11 ⊢ ( +no “ ({𝐵} × 𝐶)) ⊆ ran +no
77	76, 46	sstri 3993	. . . . . . . . . 10 ⊢ ( +no “ ({𝐵} × 𝐶)) ⊆ On
78		xpss12 5700	. . . . . . . . . 10 ⊢ (({𝐴} ⊆ On ∧ ( +no “ ({𝐵} × 𝐶)) ⊆ On) → ({𝐴} × ( +no “ ({𝐵} × 𝐶))) ⊆ (On × On))
79	42, 77, 78	sylancl 586	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → ({𝐴} × ( +no “ ({𝐵} × 𝐶))) ⊆ (On × On))
80	79, 31	sseqtrrdi 4025	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → ({𝐴} × ( +no “ ({𝐵} × 𝐶))) ⊆ dom +no )
81		funimassov 7610	. . . . . . . 8 ⊢ ((Fun +no ∧ ({𝐴} × ( +no “ ({𝐵} × 𝐶))) ⊆ dom +no ) → (( +no “ ({𝐴} × ( +no “ ({𝐵} × 𝐶)))) ⊆ 𝑥 ↔ ∀𝑎 ∈ {𝐴}∀𝑝 ∈ ( +no “ ({𝐵} × 𝐶))(𝑎 +no 𝑝) ∈ 𝑥))
82	24, 80, 81	sylancr 587	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (( +no “ ({𝐴} × ( +no “ ({𝐵} × 𝐶)))) ⊆ 𝑥 ↔ ∀𝑎 ∈ {𝐴}∀𝑝 ∈ ( +no “ ({𝐵} × 𝐶))(𝑎 +no 𝑝) ∈ 𝑥))
83	54	ralbidv 3178	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (∀𝑝 ∈ ( +no “ ({𝐵} × 𝐶))(𝑎 +no 𝑝) ∈ 𝑥 ↔ ∀𝑝 ∈ ( +no “ ({𝐵} × 𝐶))(𝐴 +no 𝑝) ∈ 𝑥))
84	83	ralsng 4675	. . . . . . . 8 ⊢ (𝐴 ∈ On → (∀𝑎 ∈ {𝐴}∀𝑝 ∈ ( +no “ ({𝐵} × 𝐶))(𝑎 +no 𝑝) ∈ 𝑥 ↔ ∀𝑝 ∈ ( +no “ ({𝐵} × 𝐶))(𝐴 +no 𝑝) ∈ 𝑥))
85	41, 84	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (∀𝑎 ∈ {𝐴}∀𝑝 ∈ ( +no “ ({𝐵} × 𝐶))(𝑎 +no 𝑝) ∈ 𝑥 ↔ ∀𝑝 ∈ ( +no “ ({𝐵} × 𝐶))(𝐴 +no 𝑝) ∈ 𝑥))
86		simpl2 1193	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → 𝐵 ∈ On)
87	86	snssd 4809	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → {𝐵} ⊆ On)
88		onss 7805	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ On → 𝐶 ⊆ On)
89	88	3ad2ant3 1136	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐶 ⊆ On)
90	89	adantr 480	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → 𝐶 ⊆ On)
91		xpss12 5700	. . . . . . . . . 10 ⊢ (({𝐵} ⊆ On ∧ 𝐶 ⊆ On) → ({𝐵} × 𝐶) ⊆ (On × On))
92	87, 90, 91	syl2anc 584	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → ({𝐵} × 𝐶) ⊆ (On × On))
93	65	imaeqalov 7672	. . . . . . . . 9 ⊢ (( +no Fn (On × On) ∧ ({𝐵} × 𝐶) ⊆ (On × On)) → (∀𝑝 ∈ ( +no “ ({𝐵} × 𝐶))(𝐴 +no 𝑝) ∈ 𝑥 ↔ ∀𝑏 ∈ {𝐵}∀𝑐 ∈ 𝐶 (𝐴 +no (𝑏 +no 𝑐)) ∈ 𝑥))
94	22, 92, 93	sylancr 587	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (∀𝑝 ∈ ( +no “ ({𝐵} × 𝐶))(𝐴 +no 𝑝) ∈ 𝑥 ↔ ∀𝑏 ∈ {𝐵}∀𝑐 ∈ 𝐶 (𝐴 +no (𝑏 +no 𝑐)) ∈ 𝑥))
95		oveq1 7438	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝐵 → (𝑏 +no 𝑐) = (𝐵 +no 𝑐))
96	95	oveq2d 7447	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝐵 → (𝐴 +no (𝑏 +no 𝑐)) = (𝐴 +no (𝐵 +no 𝑐)))
97	96	eleq1d 2826	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐵 → ((𝐴 +no (𝑏 +no 𝑐)) ∈ 𝑥 ↔ (𝐴 +no (𝐵 +no 𝑐)) ∈ 𝑥))
98	97	ralbidv 3178	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → (∀𝑐 ∈ 𝐶 (𝐴 +no (𝑏 +no 𝑐)) ∈ 𝑥 ↔ ∀𝑐 ∈ 𝐶 (𝐴 +no (𝐵 +no 𝑐)) ∈ 𝑥))
99	98	ralsng 4675	. . . . . . . . 9 ⊢ (𝐵 ∈ On → (∀𝑏 ∈ {𝐵}∀𝑐 ∈ 𝐶 (𝐴 +no (𝑏 +no 𝑐)) ∈ 𝑥 ↔ ∀𝑐 ∈ 𝐶 (𝐴 +no (𝐵 +no 𝑐)) ∈ 𝑥))
100	86, 99	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (∀𝑏 ∈ {𝐵}∀𝑐 ∈ 𝐶 (𝐴 +no (𝑏 +no 𝑐)) ∈ 𝑥 ↔ ∀𝑐 ∈ 𝐶 (𝐴 +no (𝐵 +no 𝑐)) ∈ 𝑥))
101	94, 100	bitrd 279	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (∀𝑝 ∈ ( +no “ ({𝐵} × 𝐶))(𝐴 +no 𝑝) ∈ 𝑥 ↔ ∀𝑐 ∈ 𝐶 (𝐴 +no (𝐵 +no 𝑐)) ∈ 𝑥))
102	82, 85, 101	3bitrd 305	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (( +no “ ({𝐴} × ( +no “ ({𝐵} × 𝐶)))) ⊆ 𝑥 ↔ ∀𝑐 ∈ 𝐶 (𝐴 +no (𝐵 +no 𝑐)) ∈ 𝑥))
103	40, 75, 102	3anbi123d 1438	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → ((( +no “ (𝐴 × {(𝐵 +no 𝐶)})) ⊆ 𝑥 ∧ ( +no “ ({𝐴} × ( +no “ (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 “ (𝐴 × {(𝐵 +no 𝐶)})) ∪ ( +no “ ({𝐴} × (( +no “ ({𝐵} × 𝐶)) ∪ ( +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 3443	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → {𝑥 ∈ On ∣ (( +no “ (𝐴 × {(𝐵 +no 𝐶)})) ∪ ( +no “ ({𝐴} × (( +no “ ({𝐵} × 𝐶)) ∪ ( +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 4951	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ∩ {𝑥 ∈ On ∣ (( +no “ (𝐴 × {(𝐵 +no 𝐶)})) ∪ ( +no “ ({𝐴} × (( +no “ ({𝐵} × 𝐶)) ∪ ( +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 2777	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 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061  {crab 3436   ∪ cun 3949   ⊆ wss 3951  {csn 4626  ∩ cint 4946   × cxp 5683  dom cdm 5685  ran crn 5686   “ cima 5688  Oncon0 6384  Fun wfun 6555   Fn wfn 6556  ⟶wf 6557  (class class class)co 7431   +no cnadd 8703

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-frecs 8306  df-nadd 8704

This theorem is referenced by:  naddass  8734