Theorem naddov2 8594

Description: Alternate expression for natural addition. (Contributed by Scott Fenton, 26-Aug-2024.)

Assertion

Ref	Expression
naddov2	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = ∩ {𝑥 ∈ On ∣ (∀𝑦 ∈ 𝐵 (𝐴 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝐴 (𝑧 +no 𝐵) ∈ 𝑥)})

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑦,𝐴   𝑧,𝐴   𝑦,𝐵   𝑧,𝐵   𝑥,𝑦   𝑥,𝑧

Proof of Theorem naddov2

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		naddov 8593	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = ∩ {𝑥 ∈ On ∣ (( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝐵})) ⊆ 𝑥)})
2		snssi 4757	. . . . . . . . 9 ⊢ (𝐴 ∈ On → {𝐴} ⊆ On)
3		onss 7718	. . . . . . . . 9 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On)
4		xpss12 5629	. . . . . . . . 9 ⊢ (({𝐴} ⊆ On ∧ 𝐵 ⊆ On) → ({𝐴} × 𝐵) ⊆ (On × On))
5	2, 3, 4	syl2an 596	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ({𝐴} × 𝐵) ⊆ (On × On))
6		naddfn 8590	. . . . . . . . 9 ⊢ +no Fn (On × On)
7	6	fndmi 6585	. . . . . . . 8 ⊢ dom +no = (On × On)
8	5, 7	sseqtrrdi 3971	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ({𝐴} × 𝐵) ⊆ dom +no )
9		fnfun 6581	. . . . . . . . 9 ⊢ ( +no Fn (On × On) → Fun +no )
10	6, 9	ax-mp 5	. . . . . . . 8 ⊢ Fun +no
11		funimassov 7523	. . . . . . . 8 ⊢ ((Fun +no ∧ ({𝐴} × 𝐵) ⊆ dom +no ) → (( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥 ↔ ∀𝑡 ∈ {𝐴}∀𝑦 ∈ 𝐵 (𝑡 +no 𝑦) ∈ 𝑥))
12	10, 11	mpan 690	. . . . . . 7 ⊢ (({𝐴} × 𝐵) ⊆ dom +no → (( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥 ↔ ∀𝑡 ∈ {𝐴}∀𝑦 ∈ 𝐵 (𝑡 +no 𝑦) ∈ 𝑥))
13	8, 12	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥 ↔ ∀𝑡 ∈ {𝐴}∀𝑦 ∈ 𝐵 (𝑡 +no 𝑦) ∈ 𝑥))
14		oveq1 7353	. . . . . . . . . 10 ⊢ (𝑡 = 𝐴 → (𝑡 +no 𝑦) = (𝐴 +no 𝑦))
15	14	eleq1d 2816	. . . . . . . . 9 ⊢ (𝑡 = 𝐴 → ((𝑡 +no 𝑦) ∈ 𝑥 ↔ (𝐴 +no 𝑦) ∈ 𝑥))
16	15	ralbidv 3155	. . . . . . . 8 ⊢ (𝑡 = 𝐴 → (∀𝑦 ∈ 𝐵 (𝑡 +no 𝑦) ∈ 𝑥 ↔ ∀𝑦 ∈ 𝐵 (𝐴 +no 𝑦) ∈ 𝑥))
17	16	ralsng 4625	. . . . . . 7 ⊢ (𝐴 ∈ On → (∀𝑡 ∈ {𝐴}∀𝑦 ∈ 𝐵 (𝑡 +no 𝑦) ∈ 𝑥 ↔ ∀𝑦 ∈ 𝐵 (𝐴 +no 𝑦) ∈ 𝑥))
18	17	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∀𝑡 ∈ {𝐴}∀𝑦 ∈ 𝐵 (𝑡 +no 𝑦) ∈ 𝑥 ↔ ∀𝑦 ∈ 𝐵 (𝐴 +no 𝑦) ∈ 𝑥))
19	13, 18	bitrd 279	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝐵 (𝐴 +no 𝑦) ∈ 𝑥))
20		onss 7718	. . . . . . . . 9 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
21		snssi 4757	. . . . . . . . 9 ⊢ (𝐵 ∈ On → {𝐵} ⊆ On)
22		xpss12 5629	. . . . . . . . 9 ⊢ ((𝐴 ⊆ On ∧ {𝐵} ⊆ On) → (𝐴 × {𝐵}) ⊆ (On × On))
23	20, 21, 22	syl2an 596	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 × {𝐵}) ⊆ (On × On))
24	23, 7	sseqtrrdi 3971	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 × {𝐵}) ⊆ dom +no )
25		funimassov 7523	. . . . . . . 8 ⊢ ((Fun +no ∧ (𝐴 × {𝐵}) ⊆ dom +no ) → (( +no “ (𝐴 × {𝐵})) ⊆ 𝑥 ↔ ∀𝑧 ∈ 𝐴 ∀𝑡 ∈ {𝐵} (𝑧 +no 𝑡) ∈ 𝑥))
26	10, 25	mpan 690	. . . . . . 7 ⊢ ((𝐴 × {𝐵}) ⊆ dom +no → (( +no “ (𝐴 × {𝐵})) ⊆ 𝑥 ↔ ∀𝑧 ∈ 𝐴 ∀𝑡 ∈ {𝐵} (𝑧 +no 𝑡) ∈ 𝑥))
27	24, 26	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (( +no “ (𝐴 × {𝐵})) ⊆ 𝑥 ↔ ∀𝑧 ∈ 𝐴 ∀𝑡 ∈ {𝐵} (𝑧 +no 𝑡) ∈ 𝑥))
28		oveq2 7354	. . . . . . . . . 10 ⊢ (𝑡 = 𝐵 → (𝑧 +no 𝑡) = (𝑧 +no 𝐵))
29	28	eleq1d 2816	. . . . . . . . 9 ⊢ (𝑡 = 𝐵 → ((𝑧 +no 𝑡) ∈ 𝑥 ↔ (𝑧 +no 𝐵) ∈ 𝑥))
30	29	ralsng 4625	. . . . . . . 8 ⊢ (𝐵 ∈ On → (∀𝑡 ∈ {𝐵} (𝑧 +no 𝑡) ∈ 𝑥 ↔ (𝑧 +no 𝐵) ∈ 𝑥))
31	30	ralbidv 3155	. . . . . . 7 ⊢ (𝐵 ∈ On → (∀𝑧 ∈ 𝐴 ∀𝑡 ∈ {𝐵} (𝑧 +no 𝑡) ∈ 𝑥 ↔ ∀𝑧 ∈ 𝐴 (𝑧 +no 𝐵) ∈ 𝑥))
32	31	adantl 481	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∀𝑧 ∈ 𝐴 ∀𝑡 ∈ {𝐵} (𝑧 +no 𝑡) ∈ 𝑥 ↔ ∀𝑧 ∈ 𝐴 (𝑧 +no 𝐵) ∈ 𝑥))
33	27, 32	bitrd 279	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (( +no “ (𝐴 × {𝐵})) ⊆ 𝑥 ↔ ∀𝑧 ∈ 𝐴 (𝑧 +no 𝐵) ∈ 𝑥))
34	19, 33	anbi12d 632	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝐵})) ⊆ 𝑥) ↔ (∀𝑦 ∈ 𝐵 (𝐴 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝐴 (𝑧 +no 𝐵) ∈ 𝑥)))
35	34	rabbidv 3402	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → {𝑥 ∈ On ∣ (( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝐵})) ⊆ 𝑥)} = {𝑥 ∈ On ∣ (∀𝑦 ∈ 𝐵 (𝐴 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝐴 (𝑧 +no 𝐵) ∈ 𝑥)})
36	35	inteqd 4900	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∩ {𝑥 ∈ On ∣ (( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝐵})) ⊆ 𝑥)} = ∩ {𝑥 ∈ On ∣ (∀𝑦 ∈ 𝐵 (𝐴 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝐴 (𝑧 +no 𝐵) ∈ 𝑥)})
37	1, 36	eqtrd 2766	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = ∩ {𝑥 ∈ On ∣ (∀𝑦 ∈ 𝐵 (𝐴 +no 𝑦) ∈ 𝑥 ∧ ∀𝑧 ∈ 𝐴 (𝑧 +no 𝐵) ∈ 𝑥)})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047  {crab 3395   ⊆ wss 3897  {csn 4573  ∩ cint 4895   × cxp 5612  dom cdm 5614   “ cima 5617  Oncon0 6306  Fun wfun 6475   Fn wfn 6476  (class class class)co 7346   +no cnadd 8580

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-frecs 8211  df-nadd 8581

This theorem is referenced by:  naddcom  8597  naddrid  8598  naddssim  8600  naddelim  8601  naddsuc2  8616  naddov4  43486  nadd1suc  43495