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Theorem iotassuni 4355
 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) {x φ}

Proof of Theorem iotassuni
StepHypRef Expression
1 iotauni 4351 . . 3 (∃!xφ → (℩xφ) = {x φ})
2 eqimss 3323 . . 3 ((℩xφ) = {x φ} → (℩xφ) {x φ})
31, 2syl 15 . 2 (∃!xφ → (℩xφ) {x φ})
4 0ss 3579 . . 3 {x φ}
5 iotanul 4354 . . . 4 ∃!xφ → (℩xφ) = )
65sseq1d 3298 . . 3 ∃!xφ → ((℩xφ) {x φ} ↔ {x φ}))
74, 6mpbiri 224 . 2 ∃!xφ → (℩xφ) {x φ})
83, 7pm2.61i 156 1 (℩xφ) {x φ}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1642  ∃!weu 2204  {cab 2339   ⊆ wss 3257  ∅c0 3550  ∪cuni 3891  ℩cio 4337 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-pr 3742  df-uni 3892  df-iota 4339 This theorem is referenced by: (None)
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