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| Mirrors > Home > MPE Home > Th. List > uniintab | Structured version Visualization version 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 4724 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦}) | |
| 2 | uniintsn 4984 | . 2 ⊢ (∪ {𝑥 ∣ 𝜑} = ∩ {𝑥 ∣ 𝜑} ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦}) | |
| 3 | 1, 2 | bitr4i 278 | 1 ⊢ (∃!𝑥𝜑 ↔ ∪ {𝑥 ∣ 𝜑} = ∩ {𝑥 ∣ 𝜑}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 = wceq 1539 ∃wex 1778 ∃!weu 2567 {cab 2713 {csn 4625 ∪ cuni 4906 ∩ cint 4945 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-sn 4626 df-pr 4628 df-uni 4907 df-int 4946 | 
| This theorem is referenced by: iotaint 6536 reuabaiotaiota 47104 | 
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