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Theorem iotassuni 6307
 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 6303 . . 3 (∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑})
2 eqimss 3974 . . 3 ((℩𝑥𝜑) = {𝑥𝜑} → (℩𝑥𝜑) ⊆ {𝑥𝜑})
31, 2syl 17 . 2 (∃!𝑥𝜑 → (℩𝑥𝜑) ⊆ {𝑥𝜑})
4 iotanul 6306 . . 3 (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) = ∅)
5 0ss 4307 . . 3 ∅ ⊆ {𝑥𝜑}
64, 5eqsstrdi 3972 . 2 (¬ ∃!𝑥𝜑 → (℩𝑥𝜑) ⊆ {𝑥𝜑})
73, 6pm2.61i 185 1 (℩𝑥𝜑) ⊆ {𝑥𝜑}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1538  ∃!weu 2631  {cab 2779   ⊆ wss 3884  ∅c0 4246  ∪ cuni 4803  ℩cio 6285 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-ral 3114  df-rex 3115  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-sn 4529  df-pr 4531  df-uni 4804  df-iota 6287 This theorem is referenced by:  bj-nuliotaALT  34470
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