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Mirrors > Home > NFE Home > Th. List > uniintab | GIF version |
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.) |
Ref | Expression |
---|---|
uniintab | ⊢ (∃!xφ ↔ ∪{x ∣ φ} = ∩{x ∣ φ}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | euabsn2 3791 | . 2 ⊢ (∃!xφ ↔ ∃y{x ∣ φ} = {y}) | |
2 | uniintsn 3963 | . 2 ⊢ (∪{x ∣ φ} = ∩{x ∣ φ} ↔ ∃y{x ∣ φ} = {y}) | |
3 | 1, 2 | bitr4i 243 | 1 ⊢ (∃!xφ ↔ ∪{x ∣ φ} = ∩{x ∣ φ}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∃wex 1541 = wceq 1642 ∃!weu 2204 {cab 2339 {csn 3737 ∪cuni 3891 ∩cint 3926 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-v 2861 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-int 3927 |
This theorem is referenced by: iotaint 4352 |
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