MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssoprab2b Structured version   Visualization version   GIF version

Theorem ssoprab2b 6910
Description: Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2b 5163. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
Assertion
Ref Expression
ssoprab2b ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} ↔ ∀𝑥𝑦𝑧(𝜑𝜓))

Proof of Theorem ssoprab2b
StepHypRef Expression
1 nfoprab1 6902 . . . 4 𝑥{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
2 nfoprab1 6902 . . . 4 𝑥{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}
31, 2nfss 3754 . . 3 𝑥{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}
4 nfoprab2 6903 . . . . 5 𝑦{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
5 nfoprab2 6903 . . . . 5 𝑦{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}
64, 5nfss 3754 . . . 4 𝑦{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}
7 nfoprab3 6904 . . . . . 6 𝑧{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
8 nfoprab3 6904 . . . . . 6 𝑧{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}
97, 8nfss 3754 . . . . 5 𝑧{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}
10 ssel 3755 . . . . . 6 ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} → (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}))
11 oprabid 6873 . . . . . 6 (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜑)
12 oprabid 6873 . . . . . 6 (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} ↔ 𝜓)
1310, 11, 123imtr3g 286 . . . . 5 ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} → (𝜑𝜓))
149, 13alrimi 2246 . . . 4 ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} → ∀𝑧(𝜑𝜓))
156, 14alrimi 2246 . . 3 ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} → ∀𝑦𝑧(𝜑𝜓))
163, 15alrimi 2246 . 2 ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} → ∀𝑥𝑦𝑧(𝜑𝜓))
17 ssoprab2 6909 . 2 (∀𝑥𝑦𝑧(𝜑𝜓) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓})
1816, 17impbii 200 1 ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} ↔ ∀𝑥𝑦𝑧(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wal 1650  wcel 2155  wss 3732  cop 4340  {coprab 6843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-oprab 6846
This theorem is referenced by:  eqoprab2b  6911
  Copyright terms: Public domain W3C validator