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Theorem snexxph 6894
Description: A case where the antecedent of snexg 4145 is not needed. The class {𝑥𝜑} is from dcextest 4540. (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 6370 . . 3 1o ∈ On
21elexi 2724 . 2 1o ∈ V
3 elsni 3578 . . . . 5 (𝑦 ∈ {{𝑥𝜑}} → 𝑦 = {𝑥𝜑})
4 vprc 4096 . . . . . . . 8 ¬ V ∈ V
5 df-v 2714 . . . . . . . . . 10 V = {𝑥𝑥 = 𝑥}
6 equid 1681 . . . . . . . . . . . 12 𝑥 = 𝑥
7 pm5.1im 172 . . . . . . . . . . . 12 (𝑥 = 𝑥 → (𝜑 → (𝑥 = 𝑥𝜑)))
86, 7ax-mp 5 . . . . . . . . . . 11 (𝜑 → (𝑥 = 𝑥𝜑))
98abbidv 2275 . . . . . . . . . 10 (𝜑 → {𝑥𝑥 = 𝑥} = {𝑥𝜑})
105, 9eqtr2id 2203 . . . . . . . . 9 (𝜑 → {𝑥𝜑} = V)
1110eleq1d 2226 . . . . . . . 8 (𝜑 → ({𝑥𝜑} ∈ V ↔ V ∈ V))
124, 11mtbiri 665 . . . . . . 7 (𝜑 → ¬ {𝑥𝜑} ∈ V)
13 19.8a 1570 . . . . . . . . 9 (𝑦 = {𝑥𝜑} → ∃𝑦 𝑦 = {𝑥𝜑})
143, 13syl 14 . . . . . . . 8 (𝑦 ∈ {{𝑥𝜑}} → ∃𝑦 𝑦 = {𝑥𝜑})
15 isset 2718 . . . . . . . 8 ({𝑥𝜑} ∈ V ↔ ∃𝑦 𝑦 = {𝑥𝜑})
1614, 15sylibr 133 . . . . . . 7 (𝑦 ∈ {{𝑥𝜑}} → {𝑥𝜑} ∈ V)
1712, 16nsyl3 616 . . . . . 6 (𝑦 ∈ {{𝑥𝜑}} → ¬ 𝜑)
18 vex 2715 . . . . . . . . . 10 𝑦 ∈ V
19 biidd 171 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝜑𝜑))
2018, 19elab 2856 . . . . . . . . 9 (𝑦 ∈ {𝑥𝜑} ↔ 𝜑)
2120notbii 658 . . . . . . . 8 𝑦 ∈ {𝑥𝜑} ↔ ¬ 𝜑)
2221biimpri 132 . . . . . . 7 𝜑 → ¬ 𝑦 ∈ {𝑥𝜑})
2322eq0rdv 3438 . . . . . 6 𝜑 → {𝑥𝜑} = ∅)
2417, 23syl 14 . . . . 5 (𝑦 ∈ {{𝑥𝜑}} → {𝑥𝜑} = ∅)
253, 24eqtrd 2190 . . . 4 (𝑦 ∈ {{𝑥𝜑}} → 𝑦 = ∅)
26 0lt1o 6387 . . . 4 ∅ ∈ 1o
2725, 26eqeltrdi 2248 . . 3 (𝑦 ∈ {{𝑥𝜑}} → 𝑦 ∈ 1o)
2827ssriv 3132 . 2 {{𝑥𝜑}} ⊆ 1o
292, 28ssexi 4102 1 {{𝑥𝜑}} ∈ V
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 104   = wceq 1335  wex 1472  wcel 2128  {cab 2143  Vcvv 2712  c0 3394  {csn 3560  Oncon0 4323  1oc1o 6356
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-uni 3773  df-tr 4063  df-iord 4326  df-on 4328  df-suc 4331  df-1o 6363
This theorem is referenced by: (None)
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