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Theorem iotassuni 5826
 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 5822 . . 3 (∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})
2 eqimss 3636 . . 3 ((℩𝑥𝜑) = {𝑥𝜑} → (℩𝑥𝜑) ⊆ {𝑥𝜑})
31, 2syl 17 . 2 (∃!𝑥𝜑 → (℩𝑥𝜑) ⊆ {𝑥𝜑})
4 iotanul 5825 . . 3 (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
5 0ss 3944 . . 3 ∅ ⊆ {𝑥𝜑}
64, 5syl6eqss 3634 . 2 (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) ⊆ {𝑥𝜑})
73, 6pm2.61i 176 1 (℩𝑥𝜑) ⊆ {𝑥𝜑}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1480  ∃!weu 2469  {cab 2607   ⊆ wss 3555  ∅c0 3891  ∪ cuni 4402  ℩cio 5808 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-sn 4149  df-pr 4151  df-uni 4403  df-iota 5810 This theorem is referenced by:  bj-nuliotaALT  32667
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