MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  phplem1 Structured version   Visualization version   GIF version

Theorem phplem1 8316
Description: Lemma for Pigeonhole Principle. If we join a natural number to itself minus an element, we end up with its successor minus the same element. (Contributed by NM, 25-May-1998.)
Assertion
Ref Expression
phplem1 ((𝐴 ∈ ω ∧ 𝐵𝐴) → ({𝐴} ∪ (𝐴 ∖ {𝐵})) = (suc 𝐴 ∖ {𝐵}))

Proof of Theorem phplem1
StepHypRef Expression
1 nnord 7241 . . 3 (𝐴 ∈ ω → Ord 𝐴)
2 nordeq 5896 . . . 4 ((Ord 𝐴𝐵𝐴) → 𝐴𝐵)
3 disjsn2 4395 . . . 4 (𝐴𝐵 → ({𝐴} ∩ {𝐵}) = ∅)
42, 3syl 17 . . 3 ((Ord 𝐴𝐵𝐴) → ({𝐴} ∩ {𝐵}) = ∅)
51, 4sylan 570 . 2 ((𝐴 ∈ ω ∧ 𝐵𝐴) → ({𝐴} ∩ {𝐵}) = ∅)
6 undif4 4187 . . 3 (({𝐴} ∩ {𝐵}) = ∅ → ({𝐴} ∪ (𝐴 ∖ {𝐵})) = (({𝐴} ∪ 𝐴) ∖ {𝐵}))
7 df-suc 5883 . . . . 5 suc 𝐴 = (𝐴 ∪ {𝐴})
87equncomi 3917 . . . 4 suc 𝐴 = ({𝐴} ∪ 𝐴)
98difeq1i 3882 . . 3 (suc 𝐴 ∖ {𝐵}) = (({𝐴} ∪ 𝐴) ∖ {𝐵})
106, 9syl6eqr 2826 . 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 383   = wceq 1634  wcel 2148  wne 2946  cdif 3726  cun 3727  cin 3728  c0 4073  {csn 4326  Ord word 5876  suc csuc 5879  ωcom 7233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-8 2150  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-sep 4928  ax-nul 4936  ax-pr 5048  ax-un 7117
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3or 1099  df-3an 1100  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ne 2947  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3357  df-sbc 3594  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-pss 3745  df-nul 4074  df-if 4236  df-sn 4327  df-pr 4329  df-tp 4331  df-op 4333  df-uni 4586  df-br 4798  df-opab 4860  df-tr 4900  df-eprel 5176  df-po 5184  df-so 5185  df-fr 5222  df-we 5224  df-ord 5880  df-on 5881  df-lim 5882  df-suc 5883  df-om 7234
This theorem is referenced by:  phplem2  8317
  Copyright terms: Public domain W3C validator