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Theorem rext 3978
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 2577 . . . 4 𝑥 ∈ V
21snid 3429 . . 3 𝑥 ∈ {𝑥}
3 snexgOLD 3962 . . . . 5 (𝑥 ∈ V → {𝑥} ∈ V)
41, 3ax-mp 7 . . . 4 {𝑥} ∈ V
5 eleq2 2117 . . . . 5 (𝑧 = {𝑥} → (𝑥𝑧𝑥 ∈ {𝑥}))
6 eleq2 2117 . . . . 5 (𝑧 = {𝑥} → (𝑦𝑧𝑦 ∈ {𝑥}))
75, 6imbi12d 227 . . . 4 (𝑧 = {𝑥} → ((𝑥𝑧𝑦𝑧) ↔ (𝑥 ∈ {𝑥} → 𝑦 ∈ {𝑥})))
84, 7spcv 2663 . . 3 (∀𝑧(𝑥𝑧𝑦𝑧) → (𝑥 ∈ {𝑥} → 𝑦 ∈ {𝑥}))
92, 8mpi 15 . 2 (∀𝑧(𝑥𝑧𝑦𝑧) → 𝑦 ∈ {𝑥})
10 velsn 3419 . . 3 (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
11 equcomi 1608 . . 3 (𝑦 = 𝑥𝑥 = 𝑦)
1210, 11sylbi 118 . 2 (𝑦 ∈ {𝑥} → 𝑥 = 𝑦)
139, 12syl 14 1 (∀𝑧(𝑥𝑧𝑦𝑧) → 𝑥 = 𝑦)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1257   = wceq 1259  wcel 1409  Vcvv 2574  {csn 3402
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408
This theorem is referenced by: (None)
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