Theorem elpredim 6154

Description: Membership in a predecessor class - implicative version. (Contributed by Scott Fenton, 9-May-2012.)

Hypothesis

Ref	Expression
elpredim.1	⊢ 𝑋 ∈ V

Assertion

Ref	Expression
elpredim	⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝑅𝑋)

Proof of Theorem elpredim

Step	Hyp	Ref	Expression
1		df-pred 6142	. . 3 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
2	1	elin2 4173	. 2 ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌 ∈ (◡𝑅 “ {𝑋})))
3		elpredim.1	. . . . 5 ⊢ 𝑋 ∈ V
4		elimasng 5949	. . . . . 6 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ (◡𝑅 “ {𝑋})) → (𝑌 ∈ (◡𝑅 “ {𝑋}) ↔ ⟨𝑋, 𝑌⟩ ∈ ◡𝑅))
5		opelcnvg 5745	. . . . . 6 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ (◡𝑅 “ {𝑋})) → (⟨𝑋, 𝑌⟩ ∈ ◡𝑅 ↔ ⟨𝑌, 𝑋⟩ ∈ 𝑅))
6	4, 5	bitrd 281	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ (◡𝑅 “ {𝑋})) → (𝑌 ∈ (◡𝑅 “ {𝑋}) ↔ ⟨𝑌, 𝑋⟩ ∈ 𝑅))
7	3, 6	mpan 688	. . . 4 ⊢ (𝑌 ∈ (◡𝑅 “ {𝑋}) → (𝑌 ∈ (◡𝑅 “ {𝑋}) ↔ ⟨𝑌, 𝑋⟩ ∈ 𝑅))
8	7	ibi 269	. . 3 ⊢ (𝑌 ∈ (◡𝑅 “ {𝑋}) → ⟨𝑌, 𝑋⟩ ∈ 𝑅)
9		df-br 5059	. . 3 ⊢ (𝑌𝑅𝑋 ↔ ⟨𝑌, 𝑋⟩ ∈ 𝑅)
10	8, 9	sylibr 236	. 2 ⊢ (𝑌 ∈ (◡𝑅 “ {𝑋}) → 𝑌𝑅𝑋)
11	2, 10	simplbiim 507	1 ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝑅𝑋)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∈ wcel 2110  Vcvv 3494  {csn 4560  ⟨cop 4566   class class class wbr 5058  ^◡ccnv 5548   “ cima 5552  Predcpred 6141

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-br 5059  df-opab 5121  df-xp 5555  df-cnv 5557  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142

This theorem is referenced by:  predbrg  6162  preddowncl  6169  trpredrec  33072