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

Theorem iotassuni 5905
Description: The class is a subset of the union of all elements satisfying 𝜑. (Contributed by Mario Carneiro, 24-Dec-2016.)
Assertion
Ref Expression
iotassuni (℩𝑥𝜑) ⊆ {𝑥𝜑}

Proof of Theorem iotassuni
StepHypRef Expression
1 iotauni 5901 . . 3 (∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})
2 eqimss 3690 . . 3 ((℩𝑥𝜑) = {𝑥𝜑} → (℩𝑥𝜑) ⊆ {𝑥𝜑})
31, 2syl 17 . 2 (∃!𝑥𝜑 → (℩𝑥𝜑) ⊆ {𝑥𝜑})
4 iotanul 5904 . . 3 (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
5 0ss 4005 . . 3 ∅ ⊆ {𝑥𝜑}
64, 5syl6eqss 3688 . 2 (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) ⊆ {𝑥𝜑})
73, 6pm2.61i 176 1 (℩𝑥𝜑) ⊆ {𝑥𝜑}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1523  ∃!weu 2498  {cab 2637  wss 3607  c0 3948   cuni 4468  cio 5887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-sn 4211  df-pr 4213  df-uni 4469  df-iota 5889
This theorem is referenced by:  bj-nuliotaALT  33145
  Copyright terms: Public domain W3C validator