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Theorem rext 3951
 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 vex 2560 . . . 4 𝑥 ∈ V
21snid 3402 . . 3 𝑥 ∈ {𝑥}
3 snexgOLD 3935 . . . . 5 (𝑥 ∈ V → {𝑥} ∈ V)
41, 3ax-mp 7 . . . 4 {𝑥} ∈ V
5 eleq2 2101 . . . . 5 (𝑧 = {𝑥} → (𝑥𝑧𝑥 ∈ {𝑥}))
6 eleq2 2101 . . . . 5 (𝑧 = {𝑥} → (𝑦𝑧𝑦 ∈ {𝑥}))
75, 6imbi12d 223 . . . 4 (𝑧 = {𝑥} → ((𝑥𝑧𝑦𝑧) ↔ (𝑥 ∈ {𝑥} → 𝑦 ∈ {𝑥})))
84, 7spcv 2646 . . 3 (∀𝑧(𝑥𝑧𝑦𝑧) → (𝑥 ∈ {𝑥} → 𝑦 ∈ {𝑥}))
92, 8mpi 15 . 2 (∀𝑧(𝑥𝑧𝑦𝑧) → 𝑦 ∈ {𝑥})
10 velsn 3392 . . 3 (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
11 equcomi 1592 . . 3 (𝑦 = 𝑥𝑥 = 𝑦)
1210, 11sylbi 114 . 2 (𝑦 ∈ {𝑥} → 𝑥 = 𝑦)
139, 12syl 14 1 (∀𝑧(𝑥𝑧𝑦𝑧) → 𝑥 = 𝑦)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1241   = wceq 1243   ∈ wcel 1393  Vcvv 2557  {csn 3375 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381 This theorem is referenced by: (None)
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