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Theorem phplem3 8182
Description: Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus any element of the successor. (Contributed by NM, 26-May-1998.)
Hypotheses
Ref Expression
phplem2.1 𝐴 ∈ V
phplem2.2 𝐵 ∈ V
Assertion
Ref Expression
phplem3 ((𝐴 ∈ ω ∧ 𝐵 ∈ suc 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵}))

Proof of Theorem phplem3
StepHypRef Expression
1 elsuci 5829 . 2 (𝐵 ∈ suc 𝐴 → (𝐵𝐴𝐵 = 𝐴))
2 phplem2.1 . . . 4 𝐴 ∈ V
3 phplem2.2 . . . 4 𝐵 ∈ V
42, 3phplem2 8181 . . 3 ((𝐴 ∈ ω ∧ 𝐵𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵}))
52enref 8030 . . . 4 𝐴𝐴
6 nnord 7115 . . . . . 6 (𝐴 ∈ ω → Ord 𝐴)
7 orddif 5858 . . . . . 6 (Ord 𝐴𝐴 = (suc 𝐴 ∖ {𝐴}))
86, 7syl 17 . . . . 5 (𝐴 ∈ ω → 𝐴 = (suc 𝐴 ∖ {𝐴}))
9 sneq 4220 . . . . . . 7 (𝐴 = 𝐵 → {𝐴} = {𝐵})
109difeq2d 3761 . . . . . 6 (𝐴 = 𝐵 → (suc 𝐴 ∖ {𝐴}) = (suc 𝐴 ∖ {𝐵}))
1110eqcoms 2659 . . . . 5 (𝐵 = 𝐴 → (suc 𝐴 ∖ {𝐴}) = (suc 𝐴 ∖ {𝐵}))
128, 11sylan9eq 2705 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 = 𝐴) → 𝐴 = (suc 𝐴 ∖ {𝐵}))
135, 12syl5breq 4722 . . 3 ((𝐴 ∈ ω ∧ 𝐵 = 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵}))
144, 13jaodan 843 . 2 ((𝐴 ∈ ω ∧ (𝐵𝐴𝐵 = 𝐴)) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵}))
151, 14sylan2 490 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ suc 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  cdif 3604  {csn 4210   class class class wbr 4685  Ord word 5760  suc csuc 5763  ωcom 7107  cen 7994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-om 7108  df-en 7998
This theorem is referenced by:  phplem4  8183  php  8185
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