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Theorem snexxph 6638
Description: A case where the antecedent of snexg 4010 is not needed. The class {𝑥𝜑} is from dcextest 4386. (Contributed by Mario Carneiro and Jim Kingdon, 4-Jul-2022.)
Assertion
Ref Expression
snexxph {{𝑥𝜑}} ∈ V
Distinct variable group:   𝜑,𝑥

Proof of Theorem snexxph
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 1on 6170 . . 3 1𝑜 ∈ On
21elexi 2631 . 2 1𝑜 ∈ V
3 elsni 3459 . . . . 5 (𝑦 ∈ {{𝑥𝜑}} → 𝑦 = {𝑥𝜑})
4 vprc 3963 . . . . . . . 8 ¬ V ∈ V
5 df-v 2621 . . . . . . . . . 10 V = {𝑥𝑥 = 𝑥}
6 equid 1634 . . . . . . . . . . . 12 𝑥 = 𝑥
7 pm5.1im 171 . . . . . . . . . . . 12 (𝑥 = 𝑥 → (𝜑 → (𝑥 = 𝑥𝜑)))
86, 7ax-mp 7 . . . . . . . . . . 11 (𝜑 → (𝑥 = 𝑥𝜑))
98abbidv 2205 . . . . . . . . . 10 (𝜑 → {𝑥𝑥 = 𝑥} = {𝑥𝜑})
105, 9syl5req 2133 . . . . . . . . 9 (𝜑 → {𝑥𝜑} = V)
1110eleq1d 2156 . . . . . . . 8 (𝜑 → ({𝑥𝜑} ∈ V ↔ V ∈ V))
124, 11mtbiri 635 . . . . . . 7 (𝜑 → ¬ {𝑥𝜑} ∈ V)
13 19.8a 1527 . . . . . . . . 9 (𝑦 = {𝑥𝜑} → ∃𝑦 𝑦 = {𝑥𝜑})
143, 13syl 14 . . . . . . . 8 (𝑦 ∈ {{𝑥𝜑}} → ∃𝑦 𝑦 = {𝑥𝜑})
15 isset 2625 . . . . . . . 8 ({𝑥𝜑} ∈ V ↔ ∃𝑦 𝑦 = {𝑥𝜑})
1614, 15sylibr 132 . . . . . . 7 (𝑦 ∈ {{𝑥𝜑}} → {𝑥𝜑} ∈ V)
1712, 16nsyl3 591 . . . . . 6 (𝑦 ∈ {{𝑥𝜑}} → ¬ 𝜑)
18 vex 2622 . . . . . . . . . 10 𝑦 ∈ V
19 biidd 170 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝜑𝜑))
2018, 19elab 2758 . . . . . . . . 9 (𝑦 ∈ {𝑥𝜑} ↔ 𝜑)
2120notbii 629 . . . . . . . 8 𝑦 ∈ {𝑥𝜑} ↔ ¬ 𝜑)
2221biimpri 131 . . . . . . 7 𝜑 → ¬ 𝑦 ∈ {𝑥𝜑})
2322eq0rdv 3324 . . . . . 6 𝜑 → {𝑥𝜑} = ∅)
2417, 23syl 14 . . . . 5 (𝑦 ∈ {{𝑥𝜑}} → {𝑥𝜑} = ∅)
253, 24eqtrd 2120 . . . 4 (𝑦 ∈ {{𝑥𝜑}} → 𝑦 = ∅)
26 0lt1o 6186 . . . 4 ∅ ∈ 1𝑜
2725, 26syl6eqel 2178 . . 3 (𝑦 ∈ {{𝑥𝜑}} → 𝑦 ∈ 1𝑜)
2827ssriv 3027 . 2 {{𝑥𝜑}} ⊆ 1𝑜
292, 28ssexi 3969 1 {{𝑥𝜑}} ∈ V
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 103   = wceq 1289  wex 1426  wcel 1438  {cab 2074  Vcvv 2619  c0 3284  {csn 3441  Oncon0 4181  1𝑜c1o 6156
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-uni 3649  df-tr 3929  df-iord 4184  df-on 4186  df-suc 4189  df-1o 6163
This theorem is referenced by: (None)
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