MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iotassuniOLD Structured version   Visualization version   GIF version

Theorem iotassuniOLD 6519
Description: Obsolete version of iotassuni 6512 as of 23-Dec-2024. (Contributed by Mario Carneiro, 24-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
iotassuniOLD (℩𝑥𝜑) ⊆ {𝑥𝜑}

Proof of Theorem iotassuniOLD
StepHypRef Expression
1 iotauni 6515 . . 3 (∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})
2 eqimss 4039 . . 3 ((℩𝑥𝜑) = {𝑥𝜑} → (℩𝑥𝜑) ⊆ {𝑥𝜑})
31, 2syl 17 . 2 (∃!𝑥𝜑 → (℩𝑥𝜑) ⊆ {𝑥𝜑})
4 iotanul 6518 . . 3 (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
5 0ss 4395 . . 3 ∅ ⊆ {𝑥𝜑}
64, 5eqsstrdi 4035 . 2 (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) ⊆ {𝑥𝜑})
73, 6pm2.61i 182 1 (℩𝑥𝜑) ⊆ {𝑥𝜑}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1542  ∃!weu 2563  {cab 2710  wss 3947  c0 4321   cuni 4907  cio 6490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-sn 4628  df-pr 4630  df-uni 4908  df-iota 6492
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator