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Theorem snexxph 6806
Description: A case where the antecedent of snexg 4078 is not needed. The class {𝑥𝜑} is from dcextest 4465. (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 6288 . . 3 1o ∈ On
21elexi 2672 . 2 1o ∈ V
3 elsni 3515 . . . . 5 (𝑦 ∈ {{𝑥𝜑}} → 𝑦 = {𝑥𝜑})
4 vprc 4030 . . . . . . . 8 ¬ V ∈ V
5 df-v 2662 . . . . . . . . . 10 V = {𝑥𝑥 = 𝑥}
6 equid 1662 . . . . . . . . . . . 12 𝑥 = 𝑥
7 pm5.1im 172 . . . . . . . . . . . 12 (𝑥 = 𝑥 → (𝜑 → (𝑥 = 𝑥𝜑)))
86, 7ax-mp 5 . . . . . . . . . . 11 (𝜑 → (𝑥 = 𝑥𝜑))
98abbidv 2235 . . . . . . . . . 10 (𝜑 → {𝑥𝑥 = 𝑥} = {𝑥𝜑})
105, 9syl5req 2163 . . . . . . . . 9 (𝜑 → {𝑥𝜑} = V)
1110eleq1d 2186 . . . . . . . 8 (𝜑 → ({𝑥𝜑} ∈ V ↔ V ∈ V))
124, 11mtbiri 649 . . . . . . 7 (𝜑 → ¬ {𝑥𝜑} ∈ V)
13 19.8a 1554 . . . . . . . . 9 (𝑦 = {𝑥𝜑} → ∃𝑦 𝑦 = {𝑥𝜑})
143, 13syl 14 . . . . . . . 8 (𝑦 ∈ {{𝑥𝜑}} → ∃𝑦 𝑦 = {𝑥𝜑})
15 isset 2666 . . . . . . . 8 ({𝑥𝜑} ∈ V ↔ ∃𝑦 𝑦 = {𝑥𝜑})
1614, 15sylibr 133 . . . . . . 7 (𝑦 ∈ {{𝑥𝜑}} → {𝑥𝜑} ∈ V)
1712, 16nsyl3 600 . . . . . 6 (𝑦 ∈ {{𝑥𝜑}} → ¬ 𝜑)
18 vex 2663 . . . . . . . . . 10 𝑦 ∈ V
19 biidd 171 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝜑𝜑))
2018, 19elab 2802 . . . . . . . . 9 (𝑦 ∈ {𝑥𝜑} ↔ 𝜑)
2120notbii 642 . . . . . . . 8 𝑦 ∈ {𝑥𝜑} ↔ ¬ 𝜑)
2221biimpri 132 . . . . . . 7 𝜑 → ¬ 𝑦 ∈ {𝑥𝜑})
2322eq0rdv 3377 . . . . . 6 𝜑 → {𝑥𝜑} = ∅)
2417, 23syl 14 . . . . 5 (𝑦 ∈ {{𝑥𝜑}} → {𝑥𝜑} = ∅)
253, 24eqtrd 2150 . . . 4 (𝑦 ∈ {{𝑥𝜑}} → 𝑦 = ∅)
26 0lt1o 6305 . . . 4 ∅ ∈ 1o
2725, 26syl6eqel 2208 . . 3 (𝑦 ∈ {{𝑥𝜑}} → 𝑦 ∈ 1o)
2827ssriv 3071 . 2 {{𝑥𝜑}} ⊆ 1o
292, 28ssexi 4036 1 {{𝑥𝜑}} ∈ V
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 104   = wceq 1316  wex 1453  wcel 1465  {cab 2103  Vcvv 2660  c0 3333  {csn 3497  Oncon0 4255  1oc1o 6274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-uni 3707  df-tr 3997  df-iord 4258  df-on 4260  df-suc 4263  df-1o 6281
This theorem is referenced by: (None)
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