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Mirrors > Home > NFE Home > Th. List > iunid | GIF version |
Description: An indexed union of singletons recovers the index set. (Contributed by NM, 6-Sep-2005.) |
Ref | Expression |
---|---|
iunid | ⊢ ∪x ∈ A {x} = A |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-sn 3741 | . . . . 5 ⊢ {x} = {y ∣ y = x} | |
2 | equcom 1680 | . . . . . 6 ⊢ (y = x ↔ x = y) | |
3 | 2 | abbii 2465 | . . . . 5 ⊢ {y ∣ y = x} = {y ∣ x = y} |
4 | 1, 3 | eqtri 2373 | . . . 4 ⊢ {x} = {y ∣ x = y} |
5 | 4 | a1i 10 | . . 3 ⊢ (x ∈ A → {x} = {y ∣ x = y}) |
6 | 5 | iuneq2i 3987 | . 2 ⊢ ∪x ∈ A {x} = ∪x ∈ A {y ∣ x = y} |
7 | iunab 4012 | . . 3 ⊢ ∪x ∈ A {y ∣ x = y} = {y ∣ ∃x ∈ A x = y} | |
8 | risset 2661 | . . . 4 ⊢ (y ∈ A ↔ ∃x ∈ A x = y) | |
9 | 8 | abbii 2465 | . . 3 ⊢ {y ∣ y ∈ A} = {y ∣ ∃x ∈ A x = y} |
10 | abid2 2470 | . . 3 ⊢ {y ∣ y ∈ A} = A | |
11 | 7, 9, 10 | 3eqtr2i 2379 | . 2 ⊢ ∪x ∈ A {y ∣ x = y} = A |
12 | 6, 11 | eqtri 2373 | 1 ⊢ ∪x ∈ A {x} = A |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1642 ∈ wcel 1710 {cab 2339 ∃wrex 2615 {csn 3737 ∪ciun 3969 |
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-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ral 2619 df-rex 2620 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-ss 3259 df-sn 3741 df-iun 3971 |
This theorem is referenced by: iunxpconst 4819 uniqs 5984 |
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