Theorem elpredim 5692

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 5680	. . 3 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
2	1	elin2 3801	. 2 ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌 ∈ (◡𝑅 “ {𝑋})))
3		elpredim.1	. . . . 5 ⊢ 𝑋 ∈ V
4		elimasng 5491	. . . . . 6 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ (◡𝑅 “ {𝑋})) → (𝑌 ∈ (◡𝑅 “ {𝑋}) ↔ ⟨𝑋, 𝑌⟩ ∈ ◡𝑅))
5		opelcnvg 5302	. . . . . 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 4654	. . 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 384   ∈ wcel 1990  Vcvv 3200  {csn 4177  ⟨cop 4183   class class class wbr 4653  ^◡ccnv 5113   “ cima 5117  Predcpred 5679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680

This theorem is referenced by:  predbrg  5700  preddowncl  5707  trpredrec  31738