Theorem elpredim 6128

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 6116	. . 3 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
2	1	elin2 4124	. 2 ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌 ∈ (◡𝑅 “ {𝑋})))
3		elpredim.1	. . . . 5 ⊢ 𝑋 ∈ V
4		elimasng 5922	. . . . . 6 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ (◡𝑅 “ {𝑋})) → (𝑌 ∈ (◡𝑅 “ {𝑋}) ↔ ⟨𝑋, 𝑌⟩ ∈ ◡𝑅))
5		opelcnvg 5715	. . . . . 6 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ (◡𝑅 “ {𝑋})) → (⟨𝑋, 𝑌⟩ ∈ ◡𝑅 ↔ ⟨𝑌, 𝑋⟩ ∈ 𝑅))
6	4, 5	bitrd 282	. . . . 5 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ (◡𝑅 “ {𝑋})) → (𝑌 ∈ (◡𝑅 “ {𝑋}) ↔ ⟨𝑌, 𝑋⟩ ∈ 𝑅))
7	3, 6	mpan 689	. . . 4 ⊢ (𝑌 ∈ (◡𝑅 “ {𝑋}) → (𝑌 ∈ (◡𝑅 “ {𝑋}) ↔ ⟨𝑌, 𝑋⟩ ∈ 𝑅))
8	7	ibi 270	. . 3 ⊢ (𝑌 ∈ (◡𝑅 “ {𝑋}) → ⟨𝑌, 𝑋⟩ ∈ 𝑅)
9		df-br 5031	. . 3 ⊢ (𝑌𝑅𝑋 ↔ ⟨𝑌, 𝑋⟩ ∈ 𝑅)
10	8, 9	sylibr 237	. 2 ⊢ (𝑌 ∈ (◡𝑅 “ {𝑋}) → 𝑌𝑅𝑋)
11	2, 10	simplbiim 508	1 ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑌𝑅𝑋)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∈ wcel 2111  Vcvv 3441  {csn 4525  ⟨cop 4531   class class class wbr 5030  ^◡ccnv 5518   “ cima 5522  Predcpred 6115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-xp 5525  df-cnv 5527  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116

This theorem is referenced by:  predbrg  6136  preddowncl  6143  trpredrec  33190