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Mirrors > Home > NFE Home > Th. List > euabsn | GIF version |
Description: Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by NM, 22-Feb-2004.) |
Ref | Expression |
---|---|
euabsn | ⊢ (∃!xφ ↔ ∃x{x ∣ φ} = {x}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | euabsn2 3791 | . 2 ⊢ (∃!xφ ↔ ∃y{x ∣ φ} = {y}) | |
2 | nfv 1619 | . . 3 ⊢ Ⅎy{x ∣ φ} = {x} | |
3 | nfab1 2491 | . . . 4 ⊢ Ⅎx{x ∣ φ} | |
4 | 3 | nfeq1 2498 | . . 3 ⊢ Ⅎx{x ∣ φ} = {y} |
5 | sneq 3744 | . . . 4 ⊢ (x = y → {x} = {y}) | |
6 | 5 | eqeq2d 2364 | . . 3 ⊢ (x = y → ({x ∣ φ} = {x} ↔ {x ∣ φ} = {y})) |
7 | 2, 4, 6 | cbvex 1985 | . 2 ⊢ (∃x{x ∣ φ} = {x} ↔ ∃y{x ∣ φ} = {y}) |
8 | 1, 7 | bitr4i 243 | 1 ⊢ (∃!xφ ↔ ∃x{x ∣ φ} = {x}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∃wex 1541 = wceq 1642 ∃!weu 2204 {cab 2339 {csn 3737 |
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-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-sn 3741 |
This theorem is referenced by: eusn 3796 uniintsn 3963 mapsn 6026 |
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