Theorem naddasslem2 8716

Description: Lemma for naddass 8717. 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 1154	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐴 ∈ On)
2		naddcl 8697	. . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 +no 𝐶) ∈ On)
3	2	3adant1 1148	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 +no 𝐶) ∈ On)
4		intmin 4930	. . . . 5 ⊢ (𝐴 ∈ On → ∩ {𝑎 ∈ On ∣ 𝐴 ⊆ 𝑎} = 𝐴)
5	4	eqcomd 2769	. . . 4 ⊢ (𝐴 ∈ On → 𝐴 = ∩ {𝑎 ∈ On ∣ 𝐴 ⊆ 𝑎})
6	5	3ad2ant1 1151	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐴 = ∩ {𝑎 ∈ On ∣ 𝐴 ⊆ 𝑎})
7		naddov3 8701	. . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 +no 𝐶) = ∩ {𝑝 ∈ On ∣ (( +no “ ({𝐵} × 𝐶)) ∪ ( +no “ (𝐵 × {𝐶}))) ⊆ 𝑝})
8	7	3adant1 1148	. . 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 8714	. 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 1111	. . . . . 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 466	. . . . . . . 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 5723	. . . . . . . . . . 11 ⊢ ({𝐴} × (( +no “ ({𝐵} × 𝐶)) ∪ ( +no “ (𝐵 × {𝐶})))) = (({𝐴} × ( +no “ ({𝐵} × 𝐶))) ∪ ({𝐴} × ( +no “ (𝐵 × {𝐶}))))
14	13	imaeq2i 6055	. . . . . . . . . 10 ⊢ ( +no “ ({𝐴} × (( +no “ ({𝐵} × 𝐶)) ∪ ( +no “ (𝐵 × {𝐶}))))) = ( +no “ (({𝐴} × ( +no “ ({𝐵} × 𝐶))) ∪ ({𝐴} × ( +no “ (𝐵 × {𝐶})))))
15		imaundi 6146	. . . . . . . . . 10 ⊢ ( +no “ (({𝐴} × ( +no “ ({𝐵} × 𝐶))) ∪ ({𝐴} × ( +no “ (𝐵 × {𝐶}))))) = (( +no “ ({𝐴} × ( +no “ ({𝐵} × 𝐶)))) ∪ ( +no “ ({𝐴} × ( +no “ (𝐵 × {𝐶})))))
16	14, 15	eqtri 2786	. . . . . . . . 9 ⊢ ( +no “ ({𝐴} × (( +no “ ({𝐵} × 𝐶)) ∪ ( +no “ (𝐵 × {𝐶}))))) = (( +no “ ({𝐴} × ( +no “ ({𝐵} × 𝐶)))) ∪ ( +no “ ({𝐴} × ( +no “ (𝐵 × {𝐶})))))
17	16	sseq1i 3940	. . . . . . . 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 306	. . . . . . 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 635	. . . . . 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 301	. . . . 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 8695	. . . . . . . . 9 ⊢ +no Fn (On × On)
23		fnfun 6647	. . . . . . . . 9 ⊢ ( +no Fn (On × On) → Fun +no )
24	22, 23	ax-mp 5	. . . . . . . 8 ⊢ Fun +no
25		onss 7806	. . . . . . . . . . 11 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
26	25	3ad2ant1 1151	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐴 ⊆ On)
27	3	adantr 486	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (𝐵 +no 𝐶) ∈ On)
28	27	snssd 4746	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → {(𝐵 +no 𝐶)} ⊆ On)
29		xpss12 5669	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ On ∧ {(𝐵 +no 𝐶)} ⊆ On) → (𝐴 × {(𝐵 +no 𝐶)}) ⊆ (On × On))
30	26, 28, 29	syl2an2r 698	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (𝐴 × {(𝐵 +no 𝐶)}) ⊆ (On × On))
31	22	fndmi 6651	. . . . . . . . 9 ⊢ dom +no = (On × On)
32	30, 31	sseqtrrdi 3953	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (𝐴 × {(𝐵 +no 𝐶)}) ⊆ dom +no )
33		funimassov 7605	. . . . . . . 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 599	. . . . . . 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 7460	. . . . . . . . 9 ⊢ (𝐵 +no 𝐶) ∈ V
36		oveq2 7435	. . . . . . . . . 10 ⊢ (𝑝 = (𝐵 +no 𝐶) → (𝑎 +no 𝑝) = (𝑎 +no (𝐵 +no 𝐶)))
37	36	eleq1d 2848	. . . . . . . . 9 ⊢ (𝑝 = (𝐵 +no 𝐶) → ((𝑎 +no 𝑝) ∈ 𝑥 ↔ (𝑎 +no (𝐵 +no 𝐶)) ∈ 𝑥))
38	35, 37	ralsn 4641	. . . . . . . 8 ⊢ (∀𝑝 ∈ {(𝐵 +no 𝐶)} (𝑎 +no 𝑝) ∈ 𝑥 ↔ (𝑎 +no (𝐵 +no 𝐶)) ∈ 𝑥)
39	38	ralbii 3111	. . . . . . 7 ⊢ (∀𝑎 ∈ 𝐴 ∀𝑝 ∈ {(𝐵 +no 𝐶)} (𝑎 +no 𝑝) ∈ 𝑥 ↔ ∀𝑎 ∈ 𝐴 (𝑎 +no (𝐵 +no 𝐶)) ∈ 𝑥)
40	34, 39	bitrdi 290	. . . . . 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 1210	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → 𝐴 ∈ On)
42	41	snssd 4746	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → {𝐴} ⊆ On)
43		imassrn 6203	. . . . . . . . . . 11 ⊢ ( +no “ (𝐵 × {𝐶})) ⊆ ran +no
44		naddf 8702	. . . . . . . . . . . 12 ⊢ +no :(On × On)⟶On
45		frn 6725	. . . . . . . . . . . 12 ⊢ ( +no :(On × On)⟶On → ran +no ⊆ On)
46	44, 45	ax-mp 5	. . . . . . . . . . 11 ⊢ ran +no ⊆ On
47	43, 46	sstri 3921	. . . . . . . . . 10 ⊢ ( +no “ (𝐵 × {𝐶})) ⊆ On
48		xpss12 5669	. . . . . . . . . 10 ⊢ (({𝐴} ⊆ On ∧ ( +no “ (𝐵 × {𝐶})) ⊆ On) → ({𝐴} × ( +no “ (𝐵 × {𝐶}))) ⊆ (On × On))
49	42, 47, 48	sylancl 598	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → ({𝐴} × ( +no “ (𝐵 × {𝐶}))) ⊆ (On × On))
50	49, 31	sseqtrrdi 3953	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → ({𝐴} × ( +no “ (𝐵 × {𝐶}))) ⊆ dom +no )
51		funimassov 7605	. . . . . . . 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 599	. . . . . . 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 7434	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (𝑎 +no 𝑝) = (𝐴 +no 𝑝))
54	53	eleq1d 2848	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → ((𝑎 +no 𝑝) ∈ 𝑥 ↔ (𝐴 +no 𝑝) ∈ 𝑥))
55	54	ralbidv 3188	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (∀𝑝 ∈ ( +no “ (𝐵 × {𝐶}))(𝑎 +no 𝑝) ∈ 𝑥 ↔ ∀𝑝 ∈ ( +no “ (𝐵 × {𝐶}))(𝐴 +no 𝑝) ∈ 𝑥))
56	55	ralsng 4635	. . . . . . . 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 18	. . . . . . 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 7806	. . . . . . . . . . 11 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On)
59	58	3ad2ant2 1152	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐵 ⊆ On)
60		simpl3 1212	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → 𝐶 ∈ On)
61	60	snssd 4746	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → {𝐶} ⊆ On)
62		xpss12 5669	. . . . . . . . . 10 ⊢ ((𝐵 ⊆ On ∧ {𝐶} ⊆ On) → (𝐵 × {𝐶}) ⊆ (On × On))
63	59, 61, 62	syl2an2r 698	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → (𝐵 × {𝐶}) ⊆ (On × On))
64		oveq2 7435	. . . . . . . . . . 11 ⊢ (𝑝 = (𝑏 +no 𝑐) → (𝐴 +no 𝑝) = (𝐴 +no (𝑏 +no 𝑐)))
65	64	eleq1d 2848	. . . . . . . . . 10 ⊢ (𝑝 = (𝑏 +no 𝑐) → ((𝐴 +no 𝑝) ∈ 𝑥 ↔ (𝐴 +no (𝑏 +no 𝑐)) ∈ 𝑥))
66	65	imaeqalov 7667	. . . . . . . . 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 599	. . . . . . . 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 7435	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝐶 → (𝑏 +no 𝑐) = (𝑏 +no 𝐶))
69	68	oveq2d 7443	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝐶 → (𝐴 +no (𝑏 +no 𝑐)) = (𝐴 +no (𝑏 +no 𝐶)))
70	69	eleq1d 2848	. . . . . . . . . . 11 ⊢ (𝑐 = 𝐶 → ((𝐴 +no (𝑏 +no 𝑐)) ∈ 𝑥 ↔ (𝐴 +no (𝑏 +no 𝐶)) ∈ 𝑥))
71	70	ralsng 4635	. . . . . . . . . 10 ⊢ (𝐶 ∈ On → (∀𝑐 ∈ {𝐶} (𝐴 +no (𝑏 +no 𝑐)) ∈ 𝑥 ↔ (𝐴 +no (𝑏 +no 𝐶)) ∈ 𝑥))
72	60, 71	syl 18	. . . . . . . . 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 3188	. . . . . . . 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 282	. . . . . . 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 308	. . . . . 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 6203	. . . . . . . . . . 11 ⊢ ( +no “ ({𝐵} × 𝐶)) ⊆ ran +no
77	76, 46	sstri 3921	. . . . . . . . . 10 ⊢ ( +no “ ({𝐵} × 𝐶)) ⊆ On
78		xpss12 5669	. . . . . . . . . 10 ⊢ (({𝐴} ⊆ On ∧ ( +no “ ({𝐵} × 𝐶)) ⊆ On) → ({𝐴} × ( +no “ ({𝐵} × 𝐶))) ⊆ (On × On))
79	42, 77, 78	sylancl 598	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → ({𝐴} × ( +no “ ({𝐵} × 𝐶))) ⊆ (On × On))
80	79, 31	sseqtrrdi 3953	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → ({𝐴} × ( +no “ ({𝐵} × 𝐶))) ⊆ dom +no )
81		funimassov 7605	. . . . . . . 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 599	. . . . . . 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 3188	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (∀𝑝 ∈ ( +no “ ({𝐵} × 𝐶))(𝑎 +no 𝑝) ∈ 𝑥 ↔ ∀𝑝 ∈ ( +no “ ({𝐵} × 𝐶))(𝐴 +no 𝑝) ∈ 𝑥))
84	83	ralsng 4635	. . . . . . . 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 18	. . . . . . 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 1211	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → 𝐵 ∈ On)
87	86	snssd 4746	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → {𝐵} ⊆ On)
88		onss 7806	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ On → 𝐶 ⊆ On)
89	88	3ad2ant3 1153	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐶 ⊆ On)
90	89	adantr 486	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → 𝐶 ⊆ On)
91		xpss12 5669	. . . . . . . . . 10 ⊢ (({𝐵} ⊆ On ∧ 𝐶 ⊆ On) → ({𝐵} × 𝐶) ⊆ (On × On))
92	87, 90, 91	syl2anc 596	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ 𝑥 ∈ On) → ({𝐵} × 𝐶) ⊆ (On × On))
93	65	imaeqalov 7667	. . . . . . . . 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 599	. . . . . . . 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 7434	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝐵 → (𝑏 +no 𝑐) = (𝐵 +no 𝑐))
96	95	oveq2d 7443	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝐵 → (𝐴 +no (𝑏 +no 𝑐)) = (𝐴 +no (𝐵 +no 𝑐)))
97	96	eleq1d 2848	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐵 → ((𝐴 +no (𝑏 +no 𝑐)) ∈ 𝑥 ↔ (𝐴 +no (𝐵 +no 𝑐)) ∈ 𝑥))
98	97	ralbidv 3188	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → (∀𝑐 ∈ 𝐶 (𝐴 +no (𝑏 +no 𝑐)) ∈ 𝑥 ↔ ∀𝑐 ∈ 𝐶 (𝐴 +no (𝐵 +no 𝑐)) ∈ 𝑥))
99	98	ralsng 4635	. . . . . . . . 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 18	. . . . . . . 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 282	. . . . . . 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 308	. . . . . 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 1464	. . . . 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 286	. . . 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 3421	. . 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 4914	. 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 2798	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)})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2146  ∀wral 3079  {crab 3415   ⊆ wss 3899   ∪ cun 4151  {csn 4587  ∩ cint 4909   × cxp 5652  dom cdm 5654  ran crn 5655   “ cima 5657  Oncon0 6368  Fun wfun 6540   Fn wfn 6541  ⟶wf 6542  (class class class)co 7427   +no cnadd 8685

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5234  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  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 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-pss 4016  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-int 4910  df-iun 4955  df-br 5106  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5549  df-eprel 5554  df-po 5562  df-so 5563  df-fr 5607  df-se 5608  df-we 5609  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6310  df-ord 6371  df-on 6372  df-suc 6374  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-ov 7430  df-oprab 7431  df-mpo 7432  df-1st 8008  df-2nd 8009  df-frecs 8310  df-nadd 8686

This theorem is used by:  naddass  8717