MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  phplem3OLD Structured version   Visualization version   GIF version

Theorem phplem3OLD 9256
Description: Obsolete version of phplem1 9244 as of 4-Nov-2024. (Contributed by NM, 26-May-1998.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
phplem2OLD.1 𝐴 ∈ V
phplem2OLD.2 𝐵 ∈ V
Assertion
Ref Expression
phplem3OLD ((𝐴 ∈ ω ∧ 𝐵 ∈ suc 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵}))

Proof of Theorem phplem3OLD
StepHypRef Expression
1 elsuci 6451 . 2 (𝐵 ∈ suc 𝐴 → (𝐵𝐴𝐵 = 𝐴))
2 phplem2OLD.1 . . . 4 𝐴 ∈ V
3 phplem2OLD.2 . . . 4 𝐵 ∈ V
42, 3phplem2OLD 9255 . . 3 ((𝐴 ∈ ω ∧ 𝐵𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵}))
52enref 9025 . . . 4 𝐴𝐴
6 nnord 7895 . . . . . 6 (𝐴 ∈ ω → Ord 𝐴)
7 orddif 6480 . . . . . 6 (Ord 𝐴𝐴 = (suc 𝐴 ∖ {𝐴}))
86, 7syl 17 . . . . 5 (𝐴 ∈ ω → 𝐴 = (suc 𝐴 ∖ {𝐴}))
9 sneq 4636 . . . . . . 7 (𝐴 = 𝐵 → {𝐴} = {𝐵})
109difeq2d 4126 . . . . . 6 (𝐴 = 𝐵 → (suc 𝐴 ∖ {𝐴}) = (suc 𝐴 ∖ {𝐵}))
1110eqcoms 2745 . . . . 5 (𝐵 = 𝐴 → (suc 𝐴 ∖ {𝐴}) = (suc 𝐴 ∖ {𝐵}))
128, 11sylan9eq 2797 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 = 𝐴) → 𝐴 = (suc 𝐴 ∖ {𝐵}))
135, 12breqtrid 5180 . . 3 ((𝐴 ∈ ω ∧ 𝐵 = 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵}))
144, 13jaodan 960 . 2 ((𝐴 ∈ ω ∧ (𝐵𝐴𝐵 = 𝐴)) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵}))
151, 14sylan2 593 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ suc 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 848   = wceq 1540  wcel 2108  Vcvv 3480  cdif 3948  {csn 4626   class class class wbr 5143  Ord word 6383  suc csuc 6386  ωcom 7887  cen 8982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-suc 6390  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-om 7888  df-en 8986
This theorem is referenced by:  phplem4OLD  9257  phpOLD  9259
  Copyright terms: Public domain W3C validator