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Theorem snexxph 6831
Description: A case where the antecedent of snexg 4103 is not needed. The class {𝑥𝜑} is from dcextest 4490. (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 6313 . . 3 1o ∈ On
21elexi 2693 . 2 1o ∈ V
3 elsni 3540 . . . . 5 (𝑦 ∈ {{𝑥𝜑}} → 𝑦 = {𝑥𝜑})
4 vprc 4055 . . . . . . . 8 ¬ V ∈ V
5 df-v 2683 . . . . . . . . . 10 V = {𝑥𝑥 = 𝑥}
6 equid 1677 . . . . . . . . . . . 12 𝑥 = 𝑥
7 pm5.1im 172 . . . . . . . . . . . 12 (𝑥 = 𝑥 → (𝜑 → (𝑥 = 𝑥𝜑)))
86, 7ax-mp 5 . . . . . . . . . . 11 (𝜑 → (𝑥 = 𝑥𝜑))
98abbidv 2255 . . . . . . . . . 10 (𝜑 → {𝑥𝑥 = 𝑥} = {𝑥𝜑})
105, 9syl5req 2183 . . . . . . . . 9 (𝜑 → {𝑥𝜑} = V)
1110eleq1d 2206 . . . . . . . 8 (𝜑 → ({𝑥𝜑} ∈ V ↔ V ∈ V))
124, 11mtbiri 664 . . . . . . 7 (𝜑 → ¬ {𝑥𝜑} ∈ V)
13 19.8a 1569 . . . . . . . . 9 (𝑦 = {𝑥𝜑} → ∃𝑦 𝑦 = {𝑥𝜑})
143, 13syl 14 . . . . . . . 8 (𝑦 ∈ {{𝑥𝜑}} → ∃𝑦 𝑦 = {𝑥𝜑})
15 isset 2687 . . . . . . . 8 ({𝑥𝜑} ∈ V ↔ ∃𝑦 𝑦 = {𝑥𝜑})
1614, 15sylibr 133 . . . . . . 7 (𝑦 ∈ {{𝑥𝜑}} → {𝑥𝜑} ∈ V)
1712, 16nsyl3 615 . . . . . 6 (𝑦 ∈ {{𝑥𝜑}} → ¬ 𝜑)
18 vex 2684 . . . . . . . . . 10 𝑦 ∈ V
19 biidd 171 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝜑𝜑))
2018, 19elab 2823 . . . . . . . . 9 (𝑦 ∈ {𝑥𝜑} ↔ 𝜑)
2120notbii 657 . . . . . . . 8 𝑦 ∈ {𝑥𝜑} ↔ ¬ 𝜑)
2221biimpri 132 . . . . . . 7 𝜑 → ¬ 𝑦 ∈ {𝑥𝜑})
2322eq0rdv 3402 . . . . . 6 𝜑 → {𝑥𝜑} = ∅)
2417, 23syl 14 . . . . 5 (𝑦 ∈ {{𝑥𝜑}} → {𝑥𝜑} = ∅)
253, 24eqtrd 2170 . . . 4 (𝑦 ∈ {{𝑥𝜑}} → 𝑦 = ∅)
26 0lt1o 6330 . . . 4 ∅ ∈ 1o
2725, 26syl6eqel 2228 . . 3 (𝑦 ∈ {{𝑥𝜑}} → 𝑦 ∈ 1o)
2827ssriv 3096 . 2 {{𝑥𝜑}} ⊆ 1o
292, 28ssexi 4061 1 {{𝑥𝜑}} ∈ V
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 104   = wceq 1331  wex 1468  wcel 1480  {cab 2123  Vcvv 2681  c0 3358  {csn 3522  Oncon0 4280  1oc1o 6299
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-uni 3732  df-tr 4022  df-iord 4283  df-on 4285  df-suc 4288  df-1o 6306
This theorem is referenced by: (None)
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