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

Theorem rext 4877
Description: A theorem similar to extensionality, requiring the existence of a singleton. Exercise 8 of [TakeutiZaring] p. 16. (Contributed by NM, 10-Aug-1993.)
Assertion
Ref Expression
rext (∀𝑧(𝑥𝑧𝑦𝑧) → 𝑥 = 𝑦)
Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem rext
StepHypRef Expression
1 vsnid 4180 . . 3 𝑥 ∈ {𝑥}
2 snex 4869 . . . 4 {𝑥} ∈ V
3 eleq2 2687 . . . . 5 (𝑧 = {𝑥} → (𝑥𝑧𝑥 ∈ {𝑥}))
4 eleq2 2687 . . . . 5 (𝑧 = {𝑥} → (𝑦𝑧𝑦 ∈ {𝑥}))
53, 4imbi12d 334 . . . 4 (𝑧 = {𝑥} → ((𝑥𝑧𝑦𝑧) ↔ (𝑥 ∈ {𝑥} → 𝑦 ∈ {𝑥})))
62, 5spcv 3285 . . 3 (∀𝑧(𝑥𝑧𝑦𝑧) → (𝑥 ∈ {𝑥} → 𝑦 ∈ {𝑥}))
71, 6mpi 20 . 2 (∀𝑧(𝑥𝑧𝑦𝑧) → 𝑦 ∈ {𝑥})
8 velsn 4164 . . 3 (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
9 equcomi 1941 . . 3 (𝑦 = 𝑥𝑥 = 𝑦)
108, 9sylbi 207 . 2 (𝑦 ∈ {𝑥} → 𝑥 = 𝑦)
117, 10syl 17 1 (∀𝑧(𝑥𝑧𝑦𝑧) → 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1478   = wceq 1480  wcel 1987  {csn 4148
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 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-dif 3558  df-un 3560  df-nul 3892  df-sn 4149  df-pr 4151
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator