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Theorem rext 5072
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 4367 . . 3 𝑥 ∈ {𝑥}
2 snex 5064 . . . 4 {𝑥} ∈ V
3 eleq2 2833 . . . . 5 (𝑧 = {𝑥} → (𝑥𝑧𝑥 ∈ {𝑥}))
4 eleq2 2833 . . . . 5 (𝑧 = {𝑥} → (𝑦𝑧𝑦 ∈ {𝑥}))
53, 4imbi12d 335 . . . 4 (𝑧 = {𝑥} → ((𝑥𝑧𝑦𝑧) ↔ (𝑥 ∈ {𝑥} → 𝑦 ∈ {𝑥})))
62, 5spcv 3451 . . 3 (∀𝑧(𝑥𝑧𝑦𝑧) → (𝑥 ∈ {𝑥} → 𝑦 ∈ {𝑥}))
71, 6mpi 20 . 2 (∀𝑧(𝑥𝑧𝑦𝑧) → 𝑦 ∈ {𝑥})
8 velsn 4350 . . 3 (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
9 equcomi 2114 . . 3 (𝑦 = 𝑥𝑥 = 𝑦)
108, 9sylbi 208 . 2 (𝑦 ∈ {𝑥} → 𝑥 = 𝑦)
117, 10syl 17 1 (∀𝑧(𝑥𝑧𝑦𝑧) → 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1650   = wceq 1652  wcel 2155  {csn 4334
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-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-dif 3735  df-un 3737  df-nul 4080  df-sn 4335  df-pr 4337
This theorem is referenced by: (None)
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