Theorem predonOLD 7771

Description: Obsolete version of predon 7770 as of 16-Oct-2024. (Contributed by Scott Fenton, 27-Mar-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
predonOLD	⊢ (𝐴 ∈ On → Pred( E , On, 𝐴) = 𝐴)

Proof of Theorem predonOLD

Step	Hyp	Ref	Expression
1		predep 6325	. 2 ⊢ (𝐴 ∈ On → Pred( E , On, 𝐴) = (On ∩ 𝐴))
2		onss 7769	. . 3 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
3		sseqin2 4210	. . 3 ⊢ (𝐴 ⊆ On ↔ (On ∩ 𝐴) = 𝐴)
4	2, 3	sylib 217	. 2 ⊢ (𝐴 ∈ On → (On ∩ 𝐴) = 𝐴)
5	1, 4	eqtrd 2766	1 ⊢ (𝐴 ∈ On → Pred( E , On, 𝐴) = 𝐴)

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098   ∩ cin 3942   ⊆ wss 3943   E cep 5572  Predcpred 6293  Oncon0 6358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-tr 5259  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-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362

This theorem is referenced by: (None)