Theorem uniintab 3965

Description: The union and the intersection of a class abstraction are equal exactly when there is a unique satisfying value of φ(x). (Contributed by Mario Carneiro, 24-Dec-2016.)

Assertion

Ref	Expression
uniintab	⊢ (∃!xφ ↔ ∪{x ∣ φ} = ∩{x ∣ φ})

Proof of Theorem uniintab

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		euabsn2 3792	. 2 ⊢ (∃!xφ ↔ ∃y{x ∣ φ} = {y})
2		uniintsn 3964	. 2 ⊢ (∪{x ∣ φ} = ∩{x ∣ φ} ↔ ∃y{x ∣ φ} = {y})
3	1, 2	bitr4i 243	1 ⊢ (∃!xφ ↔ ∪{x ∣ φ} = ∩{x ∣ φ})

Syntax hints:   ↔ wb 176  ∃wex 1541   = wceq 1642  ∃!weu 2204  {cab 2339  {csn 3738  ∪cuni 3892  ∩cint 3927

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-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928

This theorem is referenced by:  iotaint  4353