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Theorem snnexOLD 7009
 Description: Obsolete proof of snnex 7008 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 4829 . . . 4 ¬ V ∈ V
2 vsnid 4242 . . . . . . . . 9 𝑧 ∈ {𝑧}
3 ax6ev 1947 . . . . . . . . . 10 𝑦 𝑦 = 𝑧
4 sneq 4220 . . . . . . . . . . 11 (𝑧 = 𝑦 → {𝑧} = {𝑦})
54equcoms 1993 . . . . . . . . . 10 (𝑦 = 𝑧 → {𝑧} = {𝑦})
63, 5eximii 1804 . . . . . . . . 9 𝑦{𝑧} = {𝑦}
7 snex 4938 . . . . . . . . . 10 {𝑧} ∈ V
8 eleq2 2719 . . . . . . . . . . 11 (𝑥 = {𝑧} → (𝑧𝑥𝑧 ∈ {𝑧}))
9 eqeq1 2655 . . . . . . . . . . . 12 (𝑥 = {𝑧} → (𝑥 = {𝑦} ↔ {𝑧} = {𝑦}))
109exbidv 1890 . . . . . . . . . . 11 (𝑥 = {𝑧} → (∃𝑦 𝑥 = {𝑦} ↔ ∃𝑦{𝑧} = {𝑦}))
118, 10anbi12d 747 . . . . . . . . . 10 (𝑥 = {𝑧} → ((𝑧𝑥 ∧ ∃𝑦 𝑥 = {𝑦}) ↔ (𝑧 ∈ {𝑧} ∧ ∃𝑦{𝑧} = {𝑦})))
127, 11spcev 3331 . . . . . . . . 9 ((𝑧 ∈ {𝑧} ∧ ∃𝑦{𝑧} = {𝑦}) → ∃𝑥(𝑧𝑥 ∧ ∃𝑦 𝑥 = {𝑦}))
132, 6, 12mp2an 708 . . . . . . . 8 𝑥(𝑧𝑥 ∧ ∃𝑦 𝑥 = {𝑦})
14 eluniab 4479 . . . . . . . 8 (𝑧 {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ↔ ∃𝑥(𝑧𝑥 ∧ ∃𝑦 𝑥 = {𝑦}))
1513, 14mpbir 221 . . . . . . 7 𝑧 {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}}
16 vex 3234 . . . . . . 7 𝑧 ∈ V
1715, 162th 254 . . . . . 6 (𝑧 {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ↔ 𝑧 ∈ V)
1817eqriv 2648 . . . . 5 {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} = V
1918eleq1i 2721 . . . 4 ( {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V ↔ V ∈ V)
201, 19mtbir 312 . . 3 ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V
21 uniexg 6997 . . 3 ({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V → {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V)
2220, 21mto 188 . 2 ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V
2322nelir 2929 1 {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1523  ∃wex 1744   ∈ wcel 2030  {cab 2637   ∉ wnel 2926  Vcvv 3231  {csn 4210  ∪ cuni 4468 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-nel 2927  df-rex 2947  df-v 3233  df-dif 3610  df-un 3612  df-nul 3949  df-sn 4211  df-pr 4213  df-uni 4469 This theorem is referenced by: (None)
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