Theorem predonOLD 7822

Description: Obsolete version of predon 7821 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 6362	. 2 ⊢ (𝐴 ∈ On → Pred( E , On, 𝐴) = (On ∩ 𝐴))
2		onss 7820	. . 3 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
3		sseqin2 4244	. . 3 ⊢ (𝐴 ⊆ On ↔ (On ∩ 𝐴) = 𝐴)
4	2, 3	sylib 218	. 2 ⊢ (𝐴 ∈ On → (On ∩ 𝐴) = 𝐴)
5	1, 4	eqtrd 2780	1 ⊢ (𝐴 ∈ On → Pred( E , On, 𝐴) = 𝐴)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108   ∩ cin 3975   ⊆ wss 3976   E cep 5598  Predcpred 6331  Oncon0 6395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-tr 5284  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399

This theorem is referenced by: (None)