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Theorem phplem3OLD 9254
Description: Obsolete version of phplem1 9242 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 6453 . 2 (𝐵 ∈ suc 𝐴 → (𝐵𝐴𝐵 = 𝐴))
2 phplem2OLD.1 . . . 4 𝐴 ∈ V
3 phplem2OLD.2 . . . 4 𝐵 ∈ V
42, 3phplem2OLD 9253 . . 3 ((𝐴 ∈ ω ∧ 𝐵𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵}))
52enref 9024 . . . 4 𝐴𝐴
6 nnord 7895 . . . . . 6 (𝐴 ∈ ω → Ord 𝐴)
7 orddif 6482 . . . . . 6 (Ord 𝐴𝐴 = (suc 𝐴 ∖ {𝐴}))
86, 7syl 17 . . . . 5 (𝐴 ∈ ω → 𝐴 = (suc 𝐴 ∖ {𝐴}))
9 sneq 4641 . . . . . . 7 (𝐴 = 𝐵 → {𝐴} = {𝐵})
109difeq2d 4136 . . . . . 6 (𝐴 = 𝐵 → (suc 𝐴 ∖ {𝐴}) = (suc 𝐴 ∖ {𝐵}))
1110eqcoms 2743 . . . . 5 (𝐵 = 𝐴 → (suc 𝐴 ∖ {𝐴}) = (suc 𝐴 ∖ {𝐵}))
128, 11sylan9eq 2795 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 = 𝐴) → 𝐴 = (suc 𝐴 ∖ {𝐵}))
135, 12breqtrid 5185 . . 3 ((𝐴 ∈ ω ∧ 𝐵 = 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵}))
144, 13jaodan 959 . 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 847   = wceq 1537  wcel 2106  Vcvv 3478  cdif 3960  {csn 4631   class class class wbr 5148  Ord word 6385  suc csuc 6388  ωcom 7887  cen 8981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-suc 6392  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-om 7888  df-en 8985
This theorem is referenced by:  phplem4OLD  9255  phpOLD  9257
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