Theorem naddov2 8700

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 8699	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = ∩ {𝑥 ∈ On ∣ (( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝐵})) ⊆ 𝑥)})
2		snssi 4790	. . . . . . . . 9 ⊢ (𝐴 ∈ On → {𝐴} ⊆ On)
3		onss 7788	. . . . . . . . 9 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On)
4		xpss12 5682	. . . . . . . . 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 8696	. . . . . . . . 9 ⊢ +no Fn (On × On)
7	6	fndmi 6653	. . . . . . . 8 ⊢ dom +no = (On × On)
8	5, 7	sseqtrrdi 4007	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ({𝐴} × 𝐵) ⊆ dom +no )
9		fnfun 6649	. . . . . . . . 9 ⊢ ( +no Fn (On × On) → Fun +no )
10	6, 9	ax-mp 5	. . . . . . . 8 ⊢ Fun +no
11		funimassov 7593	. . . . . . . 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 7421	. . . . . . . . . 10 ⊢ (𝑡 = 𝐴 → (𝑡 +no 𝑦) = (𝐴 +no 𝑦))
15	14	eleq1d 2818	. . . . . . . . 9 ⊢ (𝑡 = 𝐴 → ((𝑡 +no 𝑦) ∈ 𝑥 ↔ (𝐴 +no 𝑦) ∈ 𝑥))
16	15	ralbidv 3165	. . . . . . . 8 ⊢ (𝑡 = 𝐴 → (∀𝑦 ∈ 𝐵 (𝑡 +no 𝑦) ∈ 𝑥 ↔ ∀𝑦 ∈ 𝐵 (𝐴 +no 𝑦) ∈ 𝑥))
17	16	ralsng 4657	. . . . . . 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 7788	. . . . . . . . 9 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
21		snssi 4790	. . . . . . . . 9 ⊢ (𝐵 ∈ On → {𝐵} ⊆ On)
22		xpss12 5682	. . . . . . . . 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 4007	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 × {𝐵}) ⊆ dom +no )
25		funimassov 7593	. . . . . . . 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 7422	. . . . . . . . . 10 ⊢ (𝑡 = 𝐵 → (𝑧 +no 𝑡) = (𝑧 +no 𝐵))
29	28	eleq1d 2818	. . . . . . . . 9 ⊢ (𝑡 = 𝐵 → ((𝑧 +no 𝑡) ∈ 𝑥 ↔ (𝑧 +no 𝐵) ∈ 𝑥))
30	29	ralsng 4657	. . . . . . . 8 ⊢ (𝐵 ∈ On → (∀𝑡 ∈ {𝐵} (𝑧 +no 𝑡) ∈ 𝑥 ↔ (𝑧 +no 𝐵) ∈ 𝑥))
31	30	ralbidv 3165	. . . . . . 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 3428	. . 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 4933	. 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 2769	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 1539   ∈ wcel 2107  ∀wral 3050  {crab 3420   ⊆ wss 3933  {csn 4608  ∩ cint 4928   × cxp 5665  dom cdm 5667   “ cima 5670  Oncon0 6365  Fun wfun 6536   Fn wfn 6537  (class class class)co 7414   +no cnadd 8686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5261  ax-sep 5278  ax-nul 5288  ax-pow 5347  ax-pr 5414  ax-un 7738

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3773  df-csb 3882  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-pss 3953  df-nul 4316  df-if 4508  df-pw 4584  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-int 4929  df-iun 4975  df-br 5126  df-opab 5188  df-mpt 5208  df-tr 5242  df-id 5560  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-se 5620  df-we 5621  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6303  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6495  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7997  df-2nd 7998  df-frecs 8289  df-nadd 8687

This theorem is referenced by:  naddcom  8703  naddrid  8704  naddssim  8706  naddelim  8707  naddsuc2  8722  naddov4  43341  nadd1suc  43350