Theorem naddov4 43972

Description: Alternate expression for natural addition. (Contributed by RP, 19-Dec-2024.)

Assertion

Ref	Expression
naddov4	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = ∩ ({𝑥 ∈ On ∣ ∀𝑎 ∈ 𝐴 (𝑎 +no 𝐵) ∈ 𝑥} ∩ {𝑥 ∈ On ∣ ∀𝑏 ∈ 𝐵 (𝐴 +no 𝑏) ∈ 𝑥}))

Distinct variable groups:   𝐴,𝑎,𝑥   𝐴,𝑏,𝑥   𝐵,𝑎,𝑥   𝐵,𝑏

Proof of Theorem naddov4

Step	Hyp	Ref	Expression
1		naddov2 8653	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = ∩ {𝑥 ∈ On ∣ (∀𝑏 ∈ 𝐵 (𝐴 +no 𝑏) ∈ 𝑥 ∧ ∀𝑎 ∈ 𝐴 (𝑎 +no 𝐵) ∈ 𝑥)})
2		inrab 4271	. . . 4 ⊢ ({𝑥 ∈ On ∣ ∀𝑏 ∈ 𝐵 (𝐴 +no 𝑏) ∈ 𝑥} ∩ {𝑥 ∈ On ∣ ∀𝑎 ∈ 𝐴 (𝑎 +no 𝐵) ∈ 𝑥}) = {𝑥 ∈ On ∣ (∀𝑏 ∈ 𝐵 (𝐴 +no 𝑏) ∈ 𝑥 ∧ ∀𝑎 ∈ 𝐴 (𝑎 +no 𝐵) ∈ 𝑥)}
3		incom 4164	. . . 4 ⊢ ({𝑥 ∈ On ∣ ∀𝑏 ∈ 𝐵 (𝐴 +no 𝑏) ∈ 𝑥} ∩ {𝑥 ∈ On ∣ ∀𝑎 ∈ 𝐴 (𝑎 +no 𝐵) ∈ 𝑥}) = ({𝑥 ∈ On ∣ ∀𝑎 ∈ 𝐴 (𝑎 +no 𝐵) ∈ 𝑥} ∩ {𝑥 ∈ On ∣ ∀𝑏 ∈ 𝐵 (𝐴 +no 𝑏) ∈ 𝑥})
4	2, 3	eqtr3i 2790	. . 3 ⊢ {𝑥 ∈ On ∣ (∀𝑏 ∈ 𝐵 (𝐴 +no 𝑏) ∈ 𝑥 ∧ ∀𝑎 ∈ 𝐴 (𝑎 +no 𝐵) ∈ 𝑥)} = ({𝑥 ∈ On ∣ ∀𝑎 ∈ 𝐴 (𝑎 +no 𝐵) ∈ 𝑥} ∩ {𝑥 ∈ On ∣ ∀𝑏 ∈ 𝐵 (𝐴 +no 𝑏) ∈ 𝑥})
5	4	inteqi 4912	. 2 ⊢ ∩ {𝑥 ∈ On ∣ (∀𝑏 ∈ 𝐵 (𝐴 +no 𝑏) ∈ 𝑥 ∧ ∀𝑎 ∈ 𝐴 (𝑎 +no 𝐵) ∈ 𝑥)} = ∩ ({𝑥 ∈ On ∣ ∀𝑎 ∈ 𝐴 (𝑎 +no 𝐵) ∈ 𝑥} ∩ {𝑥 ∈ On ∣ ∀𝑏 ∈ 𝐵 (𝐴 +no 𝑏) ∈ 𝑥})
6	1, 5	eqtrdi 2816	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = ∩ ({𝑥 ∈ On ∣ ∀𝑎 ∈ 𝐴 (𝑎 +no 𝐵) ∈ 𝑥} ∩ {𝑥 ∈ On ∣ ∀𝑏 ∈ 𝐵 (𝐴 +no 𝑏) ∈ 𝑥}))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∀wral 3079  {crab 3417   ∩ cin 3906  ∩ cint 4908  Oncon0 6350  (class class class)co 7400   +no cnadd 8639

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-se 5606  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-frecs 8266  df-nadd 8640

This theorem is referenced by: (None)