Theorem eliniseg 5106

Description: Membership in an initial segment. The idiom (^◡𝐴 “ {𝐵}), meaning {𝑥 ∣ 𝑥𝐴𝐵}, is used to specify an initial segment in (for example) Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 28-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Hypothesis

Ref	Expression
eliniseg.1	⊢ 𝐶 ∈ V

Assertion

Ref	Expression
eliniseg	⊢ (𝐵 ∈ 𝑉 → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵))

Proof of Theorem eliniseg

Step	Hyp	Ref	Expression
1		eliniseg.1	. 2 ⊢ 𝐶 ∈ V
2		elimasng 5104	. . . 4 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ V) → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ ◡𝐴))
3		df-br 4089	. . . 4 ⊢ (𝐵◡𝐴𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ ◡𝐴)
4	2, 3	bitr4di 198	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ V) → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐵◡𝐴𝐶))
5		brcnvg 4911	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ V) → (𝐵◡𝐴𝐶 ↔ 𝐶𝐴𝐵))
6	4, 5	bitrd 188	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ V) → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵))
7	1, 6	mpan2 425	1 ⊢ (𝐵 ∈ 𝑉 → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2202  Vcvv 2802  {csn 3669  ⟨cop 3672   class class class wbr 4088  ^◡ccnv 4724   “ cima 4728

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-cnv 4733  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738

This theorem is referenced by:  epini  5107  iniseg  5108  dfco2a  5237  isoini  5958  pilem3  15506