Users' Mathboxes Mathbox for Steven Nguyen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ss2ab1 Structured version   Visualization version   GIF version

Theorem ss2ab1 41571
Description: Class abstractions in a subclass relationship, closed form. One direction of ss2ab 4049 using fewer axioms. (Contributed by SN, 22-Dec-2024.)
Assertion
Ref Expression
ss2ab1 (∀𝑥(𝜑𝜓) → {𝑥𝜑} ⊆ {𝑥𝜓})

Proof of Theorem ss2ab1
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 spsbim 2067 . . 3 (∀𝑥(𝜑𝜓) → ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓))
2 df-clab 2702 . . 3 (𝑡 ∈ {𝑥𝜑} ↔ [𝑡 / 𝑥]𝜑)
3 df-clab 2702 . . 3 (𝑡 ∈ {𝑥𝜓} ↔ [𝑡 / 𝑥]𝜓)
41, 2, 33imtr4g 296 . 2 (∀𝑥(𝜑𝜓) → (𝑡 ∈ {𝑥𝜑} → 𝑡 ∈ {𝑥𝜓}))
54ssrdv 3981 1 (∀𝑥(𝜑𝜓) → {𝑥𝜑} ⊆ {𝑥𝜓})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1531  [wsb 2059  wcel 2098  {cab 2701  wss 3941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-in 3948  df-ss 3958
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator