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Theorem snnexOLD 7132
Description: Obsolete proof of snnex 7131 as of 5-Dec-2021. (Contributed by NM, 10-Oct-2008.) (Proof shortened by Eric Schmidt, 7-Dec-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
snnexOLD {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V
Distinct variable group:   𝑥,𝑦

Proof of Theorem snnexOLD
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 vprc 4948 . . . 4 ¬ V ∈ V
2 vsnid 4354 . . . . . . . . 9 𝑧 ∈ {𝑧}
3 ax6ev 2056 . . . . . . . . . 10 𝑦 𝑦 = 𝑧
4 sneq 4331 . . . . . . . . . . 11 (𝑧 = 𝑦 → {𝑧} = {𝑦})
54equcoms 2102 . . . . . . . . . 10 (𝑦 = 𝑧 → {𝑧} = {𝑦})
63, 5eximii 1913 . . . . . . . . 9 𝑦{𝑧} = {𝑦}
7 snex 5057 . . . . . . . . . 10 {𝑧} ∈ V
8 eleq2 2828 . . . . . . . . . . 11 (𝑥 = {𝑧} → (𝑧𝑥𝑧 ∈ {𝑧}))
9 eqeq1 2764 . . . . . . . . . . . 12 (𝑥 = {𝑧} → (𝑥 = {𝑦} ↔ {𝑧} = {𝑦}))
109exbidv 1999 . . . . . . . . . . 11 (𝑥 = {𝑧} → (∃𝑦 𝑥 = {𝑦} ↔ ∃𝑦{𝑧} = {𝑦}))
118, 10anbi12d 749 . . . . . . . . . 10 (𝑥 = {𝑧} → ((𝑧𝑥 ∧ ∃𝑦 𝑥 = {𝑦}) ↔ (𝑧 ∈ {𝑧} ∧ ∃𝑦{𝑧} = {𝑦})))
127, 11spcev 3440 . . . . . . . . 9 ((𝑧 ∈ {𝑧} ∧ ∃𝑦{𝑧} = {𝑦}) → ∃𝑥(𝑧𝑥 ∧ ∃𝑦 𝑥 = {𝑦}))
132, 6, 12mp2an 710 . . . . . . . 8 𝑥(𝑧𝑥 ∧ ∃𝑦 𝑥 = {𝑦})
14 eluniab 4599 . . . . . . . 8 (𝑧 {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ↔ ∃𝑥(𝑧𝑥 ∧ ∃𝑦 𝑥 = {𝑦}))
1513, 14mpbir 221 . . . . . . 7 𝑧 {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}}
16 vex 3343 . . . . . . 7 𝑧 ∈ V
1715, 162th 254 . . . . . 6 (𝑧 {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ↔ 𝑧 ∈ V)
1817eqriv 2757 . . . . 5 {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} = V
1918eleq1i 2830 . . . 4 ( {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V ↔ V ∈ V)
201, 19mtbir 312 . . 3 ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V
21 uniexg 7120 . . 3 ({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V → {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V)
2220, 21mto 188 . 2 ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V
2322nelir 3038 1 {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1632  wex 1853  wcel 2139  {cab 2746  wnel 3035  Vcvv 3340  {csn 4321   cuni 4588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-nel 3036  df-rex 3056  df-v 3342  df-dif 3718  df-un 3720  df-nul 4059  df-sn 4322  df-pr 4324  df-uni 4589
This theorem is referenced by: (None)
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