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

Theorem ssoprab2b 6666
Description: Equivalence of ordered pair abstraction subclass and implication. Compare ssopab2b 4967. (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 6658 . . . 4 𝑥{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
2 nfoprab1 6658 . . . 4 𝑥{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}
31, 2nfss 3581 . . 3 𝑥{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}
4 nfoprab2 6659 . . . . 5 𝑦{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
5 nfoprab2 6659 . . . . 5 𝑦{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}
64, 5nfss 3581 . . . 4 𝑦{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}
7 nfoprab3 6660 . . . . . 6 𝑧{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
8 nfoprab3 6660 . . . . . 6 𝑧{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}
97, 8nfss 3581 . . . . 5 𝑧{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}
10 ssel 3582 . . . . . 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 6632 . . . . . 6 (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜑)
12 oprabid 6632 . . . . . 6 (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} ↔ 𝜓)
1310, 11, 123imtr3g 284 . . . . 5 ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} → (𝜑𝜓))
149, 13alrimi 2085 . . . 4 ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} → ∀𝑧(𝜑𝜓))
156, 14alrimi 2085 . . 3 ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} → ∀𝑦𝑧(𝜑𝜓))
163, 15alrimi 2085 . 2 ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} → ∀𝑥𝑦𝑧(𝜑𝜓))
17 ssoprab2 6665 . 2 (∀𝑥𝑦𝑧(𝜑𝜓) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓})
1816, 17impbii 199 1 ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} ↔ ∀𝑥𝑦𝑧(𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1478  wcel 1992  wss 3560  cop 4159  {coprab 6606
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-oprab 6609
This theorem is referenced by:  eqoprab2b  6667
  Copyright terms: Public domain W3C validator