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Theorem ackbij1lem2 9206
Description: Lemma for ackbij2 9228. (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 5878 . . . 4 suc 𝐴 = (𝐴 ∪ {𝐴})
21ineq2i 3942 . . 3 (𝐵 ∩ suc 𝐴) = (𝐵 ∩ (𝐴 ∪ {𝐴}))
3 indi 4004 . . 3 (𝐵 ∩ (𝐴 ∪ {𝐴})) = ((𝐵𝐴) ∪ (𝐵 ∩ {𝐴}))
4 uncom 3888 . . 3 ((𝐵𝐴) ∪ (𝐵 ∩ {𝐴})) = ((𝐵 ∩ {𝐴}) ∪ (𝐵𝐴))
52, 3, 43eqtri 2774 . 2 (𝐵 ∩ suc 𝐴) = ((𝐵 ∩ {𝐴}) ∪ (𝐵𝐴))
6 snssi 4472 . . . 4 (𝐴𝐵 → {𝐴} ⊆ 𝐵)
7 sseqin2 3948 . . . 4 ({𝐴} ⊆ 𝐵 ↔ (𝐵 ∩ {𝐴}) = {𝐴})
86, 7sylib 208 . . 3 (𝐴𝐵 → (𝐵 ∩ {𝐴}) = {𝐴})
98uneq1d 3897 . 2 (𝐴𝐵 → ((𝐵 ∩ {𝐴}) ∪ (𝐵𝐴)) = ({𝐴} ∪ (𝐵𝐴)))
105, 9syl5eq 2794 1 (𝐴𝐵 → (𝐵 ∩ suc 𝐴) = ({𝐴} ∪ (𝐵𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1620  wcel 2127  cun 3701  cin 3702  wss 3703  {csn 4309  suc csuc 5874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-v 3330  df-un 3708  df-in 3710  df-ss 3717  df-sn 4310  df-suc 5878
This theorem is referenced by:  ackbij1lem15  9219  ackbij1lem16  9220
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