Theorem naddov2 33432

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 33431	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = ∩ {𝑥 ∈ On ∣ (( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥 ∧ ( +no “ (𝐴 × {𝐵})) ⊆ 𝑥)})
2		snssi 4701	. . . . . . . . 9 ⊢ (𝐴 ∈ On → {𝐴} ⊆ On)
3		onss 7510	. . . . . . . . 9 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On)
4		xpss12 5543	. . . . . . . . 9 ⊢ (({𝐴} ⊆ On ∧ 𝐵 ⊆ On) → ({𝐴} × 𝐵) ⊆ (On × On))
5	2, 3, 4	syl2an 598	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ({𝐴} × 𝐵) ⊆ (On × On))
6		naddfn 33428	. . . . . . . . 9 ⊢ +no Fn (On × On)
7	6	fndmi 6442	. . . . . . . 8 ⊢ dom +no = (On × On)
8	5, 7	sseqtrrdi 3945	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ({𝐴} × 𝐵) ⊆ dom +no )
9		fnfun 6439	. . . . . . . . 9 ⊢ ( +no Fn (On × On) → Fun +no )
10	6, 9	ax-mp 5	. . . . . . . 8 ⊢ Fun +no
11		funimassov 7327	. . . . . . . 8 ⊢ ((Fun +no ∧ ({𝐴} × 𝐵) ⊆ dom +no ) → (( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥 ↔ ∀𝑡 ∈ {𝐴}∀𝑦 ∈ 𝐵 (𝑡 +no 𝑦) ∈ 𝑥))
12	10, 11	mpan 689	. . . . . . 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 7163	. . . . . . . . . 10 ⊢ (𝑡 = 𝐴 → (𝑡 +no 𝑦) = (𝐴 +no 𝑦))
15	14	eleq1d 2836	. . . . . . . . 9 ⊢ (𝑡 = 𝐴 → ((𝑡 +no 𝑦) ∈ 𝑥 ↔ (𝐴 +no 𝑦) ∈ 𝑥))
16	15	ralbidv 3126	. . . . . . . 8 ⊢ (𝑡 = 𝐴 → (∀𝑦 ∈ 𝐵 (𝑡 +no 𝑦) ∈ 𝑥 ↔ ∀𝑦 ∈ 𝐵 (𝐴 +no 𝑦) ∈ 𝑥))
17	16	ralsng 4573	. . . . . . 7 ⊢ (𝐴 ∈ On → (∀𝑡 ∈ {𝐴}∀𝑦 ∈ 𝐵 (𝑡 +no 𝑦) ∈ 𝑥 ↔ ∀𝑦 ∈ 𝐵 (𝐴 +no 𝑦) ∈ 𝑥))
18	17	adantr 484	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∀𝑡 ∈ {𝐴}∀𝑦 ∈ 𝐵 (𝑡 +no 𝑦) ∈ 𝑥 ↔ ∀𝑦 ∈ 𝐵 (𝐴 +no 𝑦) ∈ 𝑥))
19	13, 18	bitrd 282	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (( +no “ ({𝐴} × 𝐵)) ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝐵 (𝐴 +no 𝑦) ∈ 𝑥))
20		onss 7510	. . . . . . . . 9 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
21		snssi 4701	. . . . . . . . 9 ⊢ (𝐵 ∈ On → {𝐵} ⊆ On)
22		xpss12 5543	. . . . . . . . 9 ⊢ ((𝐴 ⊆ On ∧ {𝐵} ⊆ On) → (𝐴 × {𝐵}) ⊆ (On × On))
23	20, 21, 22	syl2an 598	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 × {𝐵}) ⊆ (On × On))
24	23, 7	sseqtrrdi 3945	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 × {𝐵}) ⊆ dom +no )
25		funimassov 7327	. . . . . . . 8 ⊢ ((Fun +no ∧ (𝐴 × {𝐵}) ⊆ dom +no ) → (( +no “ (𝐴 × {𝐵})) ⊆ 𝑥 ↔ ∀𝑧 ∈ 𝐴 ∀𝑡 ∈ {𝐵} (𝑧 +no 𝑡) ∈ 𝑥))
26	10, 25	mpan 689	. . . . . . 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 7164	. . . . . . . . . 10 ⊢ (𝑡 = 𝐵 → (𝑧 +no 𝑡) = (𝑧 +no 𝐵))
29	28	eleq1d 2836	. . . . . . . . 9 ⊢ (𝑡 = 𝐵 → ((𝑧 +no 𝑡) ∈ 𝑥 ↔ (𝑧 +no 𝐵) ∈ 𝑥))
30	29	ralsng 4573	. . . . . . . 8 ⊢ (𝐵 ∈ On → (∀𝑡 ∈ {𝐵} (𝑧 +no 𝑡) ∈ 𝑥 ↔ (𝑧 +no 𝐵) ∈ 𝑥))
31	30	ralbidv 3126	. . . . . . 7 ⊢ (𝐵 ∈ On → (∀𝑧 ∈ 𝐴 ∀𝑡 ∈ {𝐵} (𝑧 +no 𝑡) ∈ 𝑥 ↔ ∀𝑧 ∈ 𝐴 (𝑧 +no 𝐵) ∈ 𝑥))
32	31	adantl 485	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∀𝑧 ∈ 𝐴 ∀𝑡 ∈ {𝐵} (𝑧 +no 𝑡) ∈ 𝑥 ↔ ∀𝑧 ∈ 𝐴 (𝑧 +no 𝐵) ∈ 𝑥))
33	27, 32	bitrd 282	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (( +no “ (𝐴 × {𝐵})) ⊆ 𝑥 ↔ ∀𝑧 ∈ 𝐴 (𝑧 +no 𝐵) ∈ 𝑥))
34	19, 33	anbi12d 633	. . . 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 3392	. . 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 4846	. 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 2793	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 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  {crab 3074   ⊆ wss 3860  {csn 4525  ∩ cint 4841   × cxp 5526  dom cdm 5528   “ cima 5531  Oncon0 6174  Fun wfun 6334   Fn wfn 6335  (class class class)co 7156   +no cnadd 33422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-se 5488  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7699  df-2nd 7700  df-frecs 33393  df-nadd 33423

This theorem is referenced by:  naddcom  33433  naddid1  33434  naddssim  33435  naddelim  33436