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Theorem phplem1OLD 9217
Description: Obsolete lemma for php 9210. (Contributed by NM, 25-May-1998.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
phplem1OLD ((𝐴 ∈ ω ∧ 𝐵𝐴) → ({𝐴} ∪ (𝐴 ∖ {𝐵})) = (suc 𝐴 ∖ {𝐵}))

Proof of Theorem phplem1OLD
StepHypRef Expression
1 nnord 7863 . . 3 (𝐴 ∈ ω → Ord 𝐴)
2 nordeq 6384 . . . 4 ((Ord 𝐴𝐵𝐴) → 𝐴𝐵)
3 disjsn2 4717 . . . 4 (𝐴𝐵 → ({𝐴} ∩ {𝐵}) = ∅)
42, 3syl 17 . . 3 ((Ord 𝐴𝐵𝐴) → ({𝐴} ∩ {𝐵}) = ∅)
51, 4sylan 581 . 2 ((𝐴 ∈ ω ∧ 𝐵𝐴) → ({𝐴} ∩ {𝐵}) = ∅)
6 undif4 4467 . . 3 (({𝐴} ∩ {𝐵}) = ∅ → ({𝐴} ∪ (𝐴 ∖ {𝐵})) = (({𝐴} ∪ 𝐴) ∖ {𝐵}))
7 df-suc 6371 . . . . 5 suc 𝐴 = (𝐴 ∪ {𝐴})
87equncomi 4156 . . . 4 suc 𝐴 = ({𝐴} ∪ 𝐴)
98difeq1i 4119 . . 3 (suc 𝐴 ∖ {𝐵}) = (({𝐴} ∪ 𝐴) ∖ {𝐵})
106, 9eqtr4di 2791 . 2 (({𝐴} ∩ {𝐵}) = ∅ → ({𝐴} ∪ (𝐴 ∖ {𝐵})) = (suc 𝐴 ∖ {𝐵}))
115, 10syl 17 1 ((𝐴 ∈ ω ∧ 𝐵𝐴) → ({𝐴} ∪ (𝐴 ∖ {𝐵})) = (suc 𝐴 ∖ {𝐵}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wne 2941  cdif 3946  cun 3947  cin 3948  c0 4323  {csn 4629  Ord word 6364  suc csuc 6367  ωcom 7855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-ord 6368  df-on 6369  df-suc 6371  df-om 7856
This theorem is referenced by:  phplem2OLD  9218
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