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Theorem ackbij1lem2 10218
Description: Lemma for ackbij2 10240. (Contributed by Stefan O'Rear, 18-Nov-2014.)
Assertion
Ref Expression
ackbij1lem2 (𝐴𝐵 → (𝐵 ∩ suc 𝐴) = ({𝐴} ∪ (𝐵𝐴)))

Proof of Theorem ackbij1lem2
StepHypRef Expression
1 df-suc 6364 . . . 4 suc 𝐴 = (𝐴 ∪ {𝐴})
21ineq2i 4204 . . 3 (𝐵 ∩ suc 𝐴) = (𝐵 ∩ (𝐴 ∪ {𝐴}))
3 indi 4268 . . 3 (𝐵 ∩ (𝐴 ∪ {𝐴})) = ((𝐵𝐴) ∪ (𝐵 ∩ {𝐴}))
4 uncom 4148 . . 3 ((𝐵𝐴) ∪ (𝐵 ∩ {𝐴})) = ((𝐵 ∩ {𝐴}) ∪ (𝐵𝐴))
52, 3, 43eqtri 2758 . 2 (𝐵 ∩ suc 𝐴) = ((𝐵 ∩ {𝐴}) ∪ (𝐵𝐴))
6 snssi 4806 . . . 4 (𝐴𝐵 → {𝐴} ⊆ 𝐵)
7 sseqin2 4210 . . . 4 ({𝐴} ⊆ 𝐵 ↔ (𝐵 ∩ {𝐴}) = {𝐴})
86, 7sylib 217 . . 3 (𝐴𝐵 → (𝐵 ∩ {𝐴}) = {𝐴})
98uneq1d 4157 . 2 (𝐴𝐵 → ((𝐵 ∩ {𝐴}) ∪ (𝐵𝐴)) = ({𝐴} ∪ (𝐵𝐴)))
105, 9eqtrid 2778 1 (𝐴𝐵 → (𝐵 ∩ suc 𝐴) = ({𝐴} ∪ (𝐵𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  cun 3941  cin 3942  wss 3943  {csn 4623  suc csuc 6360
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-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-un 3948  df-in 3950  df-ss 3960  df-sn 4624  df-suc 6364
This theorem is referenced by:  ackbij1lem15  10231  ackbij1lem16  10232
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