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Theorem rext 4214
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 3624 . . 3 𝑥 ∈ {𝑥}
2 vex 2740 . . . . 5 𝑥 ∈ V
32snex 4184 . . . 4 {𝑥} ∈ V
4 eleq2 2241 . . . . 5 (𝑧 = {𝑥} → (𝑥𝑧𝑥 ∈ {𝑥}))
5 eleq2 2241 . . . . 5 (𝑧 = {𝑥} → (𝑦𝑧𝑦 ∈ {𝑥}))
64, 5imbi12d 234 . . . 4 (𝑧 = {𝑥} → ((𝑥𝑧𝑦𝑧) ↔ (𝑥 ∈ {𝑥} → 𝑦 ∈ {𝑥})))
73, 6spcv 2831 . . 3 (∀𝑧(𝑥𝑧𝑦𝑧) → (𝑥 ∈ {𝑥} → 𝑦 ∈ {𝑥}))
81, 7mpi 15 . 2 (∀𝑧(𝑥𝑧𝑦𝑧) → 𝑦 ∈ {𝑥})
9 velsn 3609 . . 3 (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
10 equcomi 1704 . . 3 (𝑦 = 𝑥𝑥 = 𝑦)
119, 10sylbi 121 . 2 (𝑦 ∈ {𝑥} → 𝑥 = 𝑦)
128, 11syl 14 1 (∀𝑧(𝑥𝑧𝑦𝑧) → 𝑥 = 𝑦)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1351   = wceq 1353  wcel 2148  {csn 3592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598
This theorem is referenced by: (None)
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