Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  clsk1indlem0 Structured version   Visualization version   GIF version

Theorem clsk1indlem0 43532
Description: The ansatz closure function (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟)) has the K0 property of preserving the nullary union. (Contributed by RP, 6-Jul-2021.)
Hypothesis
Ref Expression
clsk1indlem.k 𝐾 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))
Assertion
Ref Expression
clsk1indlem0 (𝐾‘∅) = ∅

Proof of Theorem clsk1indlem0
StepHypRef Expression
1 0elpw 5351 . 2 ∅ ∈ 𝒫 3o
2 eqeq1 2729 . . . . 5 (𝑟 = ∅ → (𝑟 = {∅} ↔ ∅ = {∅}))
3 id 22 . . . . 5 (𝑟 = ∅ → 𝑟 = ∅)
42, 3ifbieq2d 4551 . . . 4 (𝑟 = ∅ → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = if(∅ = {∅}, {∅, 1o}, ∅))
5 0nep0 5353 . . . . . . 7 ∅ ≠ {∅}
65a1i 11 . . . . . 6 (𝑟 = ∅ → ∅ ≠ {∅})
76neneqd 2935 . . . . 5 (𝑟 = ∅ → ¬ ∅ = {∅})
87iffalsed 4536 . . . 4 (𝑟 = ∅ → if(∅ = {∅}, {∅, 1o}, ∅) = ∅)
94, 8eqtrd 2765 . . 3 (𝑟 = ∅ → if(𝑟 = {∅}, {∅, 1o}, 𝑟) = ∅)
10 clsk1indlem.k . . 3 𝐾 = (𝑟 ∈ 𝒫 3o ↦ if(𝑟 = {∅}, {∅, 1o}, 𝑟))
11 0ex 5303 . . 3 ∅ ∈ V
129, 10, 11fvmpt 6998 . 2 (∅ ∈ 𝒫 3o → (𝐾‘∅) = ∅)
131, 12ax-mp 5 1 (𝐾‘∅) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wcel 2098  wne 2930  c0 4319  ifcif 4525  𝒫 cpw 4599  {csn 4625  {cpr 4627  cmpt 5227  cfv 6543  1oc1o 8473  3oc3o 8475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3944  df-un 3946  df-ss 3958  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-iota 6495  df-fun 6545  df-fv 6551
This theorem is referenced by:  clsk1independent  43537
  Copyright terms: Public domain W3C validator