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Theorem rext 5336
 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 4598 . . 3 𝑥 ∈ {𝑥}
2 snex 5327 . . . 4 {𝑥} ∈ V
3 eleq2 2905 . . . . 5 (𝑧 = {𝑥} → (𝑥𝑧𝑥 ∈ {𝑥}))
4 eleq2 2905 . . . . 5 (𝑧 = {𝑥} → (𝑦𝑧𝑦 ∈ {𝑥}))
53, 4imbi12d 346 . . . 4 (𝑧 = {𝑥} → ((𝑥𝑧𝑦𝑧) ↔ (𝑥 ∈ {𝑥} → 𝑦 ∈ {𝑥})))
62, 5spcv 3609 . . 3 (∀𝑧(𝑥𝑧𝑦𝑧) → (𝑥 ∈ {𝑥} → 𝑦 ∈ {𝑥}))
71, 6mpi 20 . 2 (∀𝑧(𝑥𝑧𝑦𝑧) → 𝑦 ∈ {𝑥})
8 velsn 4579 . . 3 (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
9 equcomi 2017 . . 3 (𝑦 = 𝑥𝑥 = 𝑦)
108, 9sylbi 218 . 2 (𝑦 ∈ {𝑥} → 𝑥 = 𝑦)
117, 10syl 17 1 (∀𝑧(𝑥𝑧𝑦𝑧) → 𝑥 = 𝑦)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1528   = wceq 1530   ∈ wcel 2107  {csn 4563 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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-v 3501  df-dif 3942  df-un 3944  df-nul 4295  df-sn 4564  df-pr 4566 This theorem is referenced by: (None)
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