Theorem naddov2 8620

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 8619	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = ∩ {𝑥 ∈ On ∣ (( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝐵})) ⊆ 𝑥)})
2		snssi 4768	. . . . . . . . 9 ⊢ (𝐴 ∈ On → {𝐴} ⊆ On)
3		onss 7741	. . . . . . . . 9 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On)
4		xpss12 5646	. . . . . . . . 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 8616	. . . . . . . . 9 ⊢ +no Fn (On × On)
7	6	fndmi 6604	. . . . . . . 8 ⊢ dom +no = (On × On)
8	5, 7	sseqtrrdi 3985	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ({𝐴} × 𝐵) ⊆ dom +no )
9		fnfun 6600	. . . . . . . . 9 ⊢ ( +no Fn (On × On) → Fun +no )
10	6, 9	ax-mp 5	. . . . . . . 8 ⊢ Fun +no
11		funimassov 7546	. . . . . . . 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 7376	. . . . . . . . . 10 ⊢ (𝑡 = 𝐴 → (𝑡 +no 𝑦) = (𝐴 +no 𝑦))
15	14	eleq1d 2813	. . . . . . . . 9 ⊢ (𝑡 = 𝐴 → ((𝑡 +no 𝑦) ∈ 𝑥 ↔ (𝐴 +no 𝑦) ∈ 𝑥))
16	15	ralbidv 3156	. . . . . . . 8 ⊢ (𝑡 = 𝐴 → (∀𝑦 ∈ 𝐵 (𝑡 +no 𝑦) ∈ 𝑥 ↔ ∀𝑦 ∈ 𝐵 (𝐴 +no 𝑦) ∈ 𝑥))
17	16	ralsng 4635	. . . . . . 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 7741	. . . . . . . . 9 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
21		snssi 4768	. . . . . . . . 9 ⊢ (𝐵 ∈ On → {𝐵} ⊆ On)
22		xpss12 5646	. . . . . . . . 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 3985	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 × {𝐵}) ⊆ dom +no )
25		funimassov 7546	. . . . . . . 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 7377	. . . . . . . . . 10 ⊢ (𝑡 = 𝐵 → (𝑧 +no 𝑡) = (𝑧 +no 𝐵))
29	28	eleq1d 2813	. . . . . . . . 9 ⊢ (𝑡 = 𝐵 → ((𝑧 +no 𝑡) ∈ 𝑥 ↔ (𝑧 +no 𝐵) ∈ 𝑥))
30	29	ralsng 4635	. . . . . . . 8 ⊢ (𝐵 ∈ On → (∀𝑡 ∈ {𝐵} (𝑧 +no 𝑡) ∈ 𝑥 ↔ (𝑧 +no 𝐵) ∈ 𝑥))
31	30	ralbidv 3156	. . . . . . 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 3410	. . 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 4911	. 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 2764	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 1540   ∈ wcel 2109  ∀wral 3044  {crab 3402   ⊆ wss 3911  {csn 4585  ∩ cint 4906   × cxp 5629  dom cdm 5631   “ cima 5634  Oncon0 6320  Fun wfun 6493   Fn wfn 6494  (class class class)co 7369   +no cnadd 8606

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 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-frecs 8237  df-nadd 8607

This theorem is referenced by:  naddcom  8623  naddrid  8624  naddssim  8626  naddelim  8627  naddsuc2  8642  naddov4  43365  nadd1suc  43374