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Theorem phplem3OLD 9202
Description: Obsolete version of phplem1 9190 as of 23-Sep-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 6420 . 2 (𝐵 ∈ suc 𝐴 → (𝐵𝐴𝐵 = 𝐴))
2 phplem2OLD.1 . . . 4 𝐴 ∈ V
3 phplem2OLD.2 . . . 4 𝐵 ∈ V
42, 3phplem2OLD 9201 . . 3 ((𝐴 ∈ ω ∧ 𝐵𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵}))
52enref 8964 . . . 4 𝐴𝐴
6 nnord 7846 . . . . . 6 (𝐴 ∈ ω → Ord 𝐴)
7 orddif 6449 . . . . . 6 (Ord 𝐴𝐴 = (suc 𝐴 ∖ {𝐴}))
86, 7syl 17 . . . . 5 (𝐴 ∈ ω → 𝐴 = (suc 𝐴 ∖ {𝐴}))
9 sneq 4632 . . . . . . 7 (𝐴 = 𝐵 → {𝐴} = {𝐵})
109difeq2d 4118 . . . . . 6 (𝐴 = 𝐵 → (suc 𝐴 ∖ {𝐴}) = (suc 𝐴 ∖ {𝐵}))
1110eqcoms 2739 . . . . 5 (𝐵 = 𝐴 → (suc 𝐴 ∖ {𝐴}) = (suc 𝐴 ∖ {𝐵}))
128, 11sylan9eq 2791 . . . 4 ((𝐴 ∈ ω ∧ 𝐵 = 𝐴) → 𝐴 = (suc 𝐴 ∖ {𝐵}))
135, 12breqtrid 5178 . . 3 ((𝐴 ∈ ω ∧ 𝐵 = 𝐴) → 𝐴 ≈ (suc 𝐴 ∖ {𝐵}))
144, 13jaodan 956 . 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 396  wo 845   = wceq 1541  wcel 2106  Vcvv 3473  cdif 3941  {csn 4622   class class class wbr 5141  Ord word 6352  suc csuc 6355  ωcom 7838  cen 8919
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6356  df-on 6357  df-suc 6359  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-om 7839  df-en 8923
This theorem is referenced by:  phplem4OLD  9203  phpOLD  9205
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