Theorem elpredim 5730

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 5718	. . 3 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
2	1	elin2 3834	. 2 ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌 ∈ (◡𝑅 “ {𝑋})))
3		elpredim.1	. . . . 5 ⊢ 𝑋 ∈ V
4		elimasng 5526	. . . . . 6 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ (◡𝑅 “ {𝑋})) → (𝑌 ∈ (◡𝑅 “ {𝑋}) ↔ ⟨𝑋, 𝑌⟩ ∈ ◡𝑅))
5		opelcnvg 5334	. . . . . 6 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ (◡𝑅 “ {𝑋})) → (⟨𝑋, 𝑌⟩ ∈ ◡𝑅 ↔ ⟨𝑌, 𝑋⟩ ∈ 𝑅))
6	4, 5	bitrd 268	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ (◡𝑅 “ {𝑋})) → (𝑌 ∈ (◡𝑅 “ {𝑋}) ↔ ⟨𝑌, 𝑋⟩ ∈ 𝑅))
7	3, 6	mpan 706	. . . 4 ⊢ (𝑌 ∈ (◡𝑅 “ {𝑋}) → (𝑌 ∈ (◡𝑅 “ {𝑋}) ↔ ⟨𝑌, 𝑋⟩ ∈ 𝑅))
8	7	ibi 256	. . 3 ⊢ (𝑌 ∈ (◡𝑅 “ {𝑋}) → ⟨𝑌, 𝑋⟩ ∈ 𝑅)
9		df-br 4686	. . 3 ⊢ (𝑌𝑅𝑋 ↔ ⟨𝑌, 𝑋⟩ ∈ 𝑅)
10	8, 9	sylibr 224	. 2 ⊢ (𝑌 ∈ (◡𝑅 “ {𝑋}) → 𝑌𝑅𝑋)
11	2, 10	simplbiim 659	1 ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝑅𝑋)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∈ wcel 2030  Vcvv 3231  {csn 4210  ⟨cop 4216   class class class wbr 4685  ^◡ccnv 5142   “ cima 5146  Predcpred 5717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718

This theorem is referenced by:  predbrg  5738  preddowncl  5745  trpredrec  31862