Theorem naddov4 44479

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 8692	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = ∩ {𝑥 ∈ On ∣ (∀𝑏 ∈ 𝐵 (𝐴 +no 𝑏) ∈ 𝑥 ∧ ∀𝑎 ∈ 𝐴 (𝑎 +no 𝐵) ∈ 𝑥)})
2		inrab 4261	. . . 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 4106	. . . 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 2785	. . 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 4910	. 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 2811	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +no 𝐵) = ∩ ({𝑥 ∈ On ∣ ∀𝑎 ∈ 𝐴 (𝑎 +no 𝐵) ∈ 𝑥} ∩ {𝑥 ∈ On ∣ ∀𝑏 ∈ 𝐵 (𝐴 +no 𝑏) ∈ 𝑥}))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  {crab 3412   ∩ cin 4089  ∩ cint 4906  Oncon0 6362  (class class class)co 7421   +no cnadd 8678

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7424  df-oprab 7425  df-mpo 7426  df-1st 8002  df-2nd 8003  df-frecs 8303  df-nadd 8679

This theorem is used by: (None)