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Theorem clublem 37736
Description: If a superset 𝑌 of 𝑋 possesses the property parameterized in 𝑥 in 𝜓, then 𝑌 is a superset of the closure of that property for the set 𝑋. (Contributed by RP, 23-Jul-2020.)
Hypotheses
Ref Expression
clublem.y (𝜑𝑌 ∈ V)
clublem.sub (𝑥 = 𝑌 → (𝜓𝜒))
clublem.sup (𝜑𝑋𝑌)
clublem.maj (𝜑𝜒)
Assertion
Ref Expression
clublem (𝜑 {𝑥 ∣ (𝑋𝑥𝜓)} ⊆ 𝑌)
Distinct variable groups:   𝜒,𝑥   𝑥,𝑋   𝑥,𝑌
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem clublem
StepHypRef Expression
1 clublem.sup . . 3 (𝜑𝑋𝑌)
2 clublem.maj . . 3 (𝜑𝜒)
3 clublem.y . . . . 5 (𝜑𝑌 ∈ V)
43a1d 25 . . . 4 (𝜑 → ((𝑋𝑌𝜒) → 𝑌 ∈ V))
5 clublem.sub . . . . . 6 (𝑥 = 𝑌 → (𝜓𝜒))
65cleq2lem 37733 . . . . 5 (𝑥 = 𝑌 → ((𝑋𝑥𝜓) ↔ (𝑋𝑌𝜒)))
76elab3g 3351 . . . 4 (((𝑋𝑌𝜒) → 𝑌 ∈ V) → (𝑌 ∈ {𝑥 ∣ (𝑋𝑥𝜓)} ↔ (𝑋𝑌𝜒)))
84, 7syl 17 . . 3 (𝜑 → (𝑌 ∈ {𝑥 ∣ (𝑋𝑥𝜓)} ↔ (𝑋𝑌𝜒)))
91, 2, 8mpbir2and 956 . 2 (𝜑𝑌 ∈ {𝑥 ∣ (𝑋𝑥𝜓)})
10 intss1 4483 . 2 (𝑌 ∈ {𝑥 ∣ (𝑋𝑥𝜓)} → {𝑥 ∣ (𝑋𝑥𝜓)} ⊆ 𝑌)
119, 10syl 17 1 (𝜑 {𝑥 ∣ (𝑋𝑥𝜓)} ⊆ 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  {cab 2606  Vcvv 3195  wss 3567   cint 4466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-v 3197  df-in 3574  df-ss 3581  df-int 4467
This theorem is referenced by:  mptrcllem  37739  trclubgNEW  37744
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