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Mirrors > Home > ILE Home > Th. List > uniintsnr | GIF version |
Description: The union and intersection of a singleton are equal. See also eusn 3435. (Contributed by Jim Kingdon, 14-Aug-2018.) |
Ref | Expression |
---|---|
uniintsnr | ⊢ (∃x A = {x} → ∪ A = ∩ A) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2554 | . . . 4 ⊢ x ∈ V | |
2 | 1 | unisn 3587 | . . 3 ⊢ ∪ {x} = x |
3 | unieq 3580 | . . 3 ⊢ (A = {x} → ∪ A = ∪ {x}) | |
4 | inteq 3609 | . . . 4 ⊢ (A = {x} → ∩ A = ∩ {x}) | |
5 | 1 | intsn 3641 | . . . 4 ⊢ ∩ {x} = x |
6 | 4, 5 | syl6eq 2085 | . . 3 ⊢ (A = {x} → ∩ A = x) |
7 | 2, 3, 6 | 3eqtr4a 2095 | . 2 ⊢ (A = {x} → ∪ A = ∩ A) |
8 | 7 | exlimiv 1486 | 1 ⊢ (∃x A = {x} → ∪ A = ∩ A) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1242 ∃wex 1378 {csn 3367 ∪ cuni 3571 ∩ cint 3606 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-sn 3373 df-pr 3374 df-uni 3572 df-int 3607 |
This theorem is referenced by: uniintabim 3643 |
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