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Theorem perfi 23178
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 23176 . . 3 (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋 ¬ {𝑥} ∈ 𝐽))
32simprbi 496 . 2 (𝐽 ∈ Perf → ∀𝑥𝑋 ¬ {𝑥} ∈ 𝐽)
4 sneq 4640 . . . . 5 (𝑥 = 𝑃 → {𝑥} = {𝑃})
54eleq1d 2823 . . . 4 (𝑥 = 𝑃 → ({𝑥} ∈ 𝐽 ↔ {𝑃} ∈ 𝐽))
65notbid 318 . . 3 (𝑥 = 𝑃 → (¬ {𝑥} ∈ 𝐽 ↔ ¬ {𝑃} ∈ 𝐽))
76rspccva 3620 . 2 ((∀𝑥𝑋 ¬ {𝑥} ∈ 𝐽𝑃𝑋) → ¬ {𝑃} ∈ 𝐽)
83, 7sylan 580 1 ((𝐽 ∈ Perf ∧ 𝑃𝑋) → ¬ {𝑃} ∈ 𝐽)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1536  wcel 2105  wral 3058  {csn 4630   cuni 4911  Topctop 22914  Perfcperf 23158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-top 22915  df-cld 23042  df-ntr 23043  df-cls 23044  df-lp 23159  df-perf 23160
This theorem is referenced by:  perfopn  23208
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