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Theorem partsuc2 38283
Description: Property of the partition. (Contributed by Peter Mazsa, 24-Jul-2024.)
Assertion
Ref Expression
partsuc2 (((𝑅 ↾ (𝐴 ∪ {𝐴})) ∖ (𝑅 ↾ {𝐴})) Part ((𝐴 ∪ {𝐴}) ∖ {𝐴}) ↔ (𝑅𝐴) Part 𝐴)

Proof of Theorem partsuc2
StepHypRef Expression
1 ressucdifsn2 37747 . 2 ((𝑅 ↾ (𝐴 ∪ {𝐴})) ∖ (𝑅 ↾ {𝐴})) = (𝑅𝐴)
2 sucdifsn2 37741 . 2 ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = 𝐴
3 parteq12 38280 . 2 ((((𝑅 ↾ (𝐴 ∪ {𝐴})) ∖ (𝑅 ↾ {𝐴})) = (𝑅𝐴) ∧ ((𝐴 ∪ {𝐴}) ∖ {𝐴}) = 𝐴) → (((𝑅 ↾ (𝐴 ∪ {𝐴})) ∖ (𝑅 ↾ {𝐴})) Part ((𝐴 ∪ {𝐴}) ∖ {𝐴}) ↔ (𝑅𝐴) Part 𝐴))
41, 2, 3mp2an 690 1 (((𝑅 ↾ (𝐴 ∪ {𝐴})) ∖ (𝑅 ↾ {𝐴})) Part ((𝐴 ∪ {𝐴}) ∖ {𝐴}) ↔ (𝑅𝐴) Part 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1533  cdif 3946  cun 3947  {csn 4632  cres 5684   Part wpart 37720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-reg 9623
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ec 8733  df-qs 8737  df-coss 37915  df-cnvrefrel 38031  df-dmqs 38143  df-funALTV 38186  df-disjALTV 38209  df-part 38270
This theorem is referenced by: (None)
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