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Theorem perfi 21159
Description: Property of a perfect space. (Contributed by Mario Carneiro, 24-Dec-2016.)
Hypothesis
Ref Expression
lpfval.1 𝑋 = 𝐽
Assertion
Ref Expression
perfi ((𝐽 ∈ Perf ∧ 𝑃𝑋) → ¬ {𝑃} ∈ 𝐽)

Proof of Theorem perfi
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 lpfval.1 . . . 4 𝑋 = 𝐽
21isperf3 21157 . . 3 (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋 ¬ {𝑥} ∈ 𝐽))
32simprbi 483 . 2 (𝐽 ∈ Perf → ∀𝑥𝑋 ¬ {𝑥} ∈ 𝐽)
4 sneq 4329 . . . . 5 (𝑥 = 𝑃 → {𝑥} = {𝑃})
54eleq1d 2822 . . . 4 (𝑥 = 𝑃 → ({𝑥} ∈ 𝐽 ↔ {𝑃} ∈ 𝐽))
65notbid 307 . . 3 (𝑥 = 𝑃 → (¬ {𝑥} ∈ 𝐽 ↔ ¬ {𝑃} ∈ 𝐽))
76rspccva 3446 . 2 ((∀𝑥𝑋 ¬ {𝑥} ∈ 𝐽𝑃𝑋) → ¬ {𝑃} ∈ 𝐽)
83, 7sylan 489 1 ((𝐽 ∈ Perf ∧ 𝑃𝑋) → ¬ {𝑃} ∈ 𝐽)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1630  wcel 2137  wral 3048  {csn 4319   cuni 4586  Topctop 20898  Perfcperf 21139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-reu 3055  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-int 4626  df-iun 4672  df-iin 4673  df-br 4803  df-opab 4863  df-mpt 4880  df-id 5172  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-top 20899  df-cld 21023  df-ntr 21024  df-cls 21025  df-lp 21140  df-perf 21141
This theorem is referenced by:  perfopn  21189
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