Theorem predonOLD 7787

Description: Obsolete version of predon 7786 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 6331	. 2 ⊢ (𝐴 ∈ On → Pred( E , On, 𝐴) = (On ∩ 𝐴))
2		onss 7785	. . 3 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
3		sseqin2 4209	. . 3 ⊢ (𝐴 ⊆ On ↔ (On ∩ 𝐴) = 𝐴)
4	2, 3	sylib 217	. 2 ⊢ (𝐴 ∈ On → (On ∩ 𝐴) = 𝐴)
5	1, 4	eqtrd 2765	1 ⊢ (𝐴 ∈ On → Pred( E , On, 𝐴) = 𝐴)

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098   ∩ cin 3938   ⊆ wss 3939   E cep 5575  Predcpred 6299  Oncon0 6364

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 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-tr 5261  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368

This theorem is referenced by: (None)