![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > euabsn | Structured version Visualization version 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 4691 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦}) | |
2 | nfv 1918 | . . 3 ⊢ Ⅎ𝑦{𝑥 ∣ 𝜑} = {𝑥} | |
3 | nfab1 2910 | . . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | |
4 | 3 | nfeq1 2923 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} = {𝑦} |
5 | sneq 4601 | . . . 4 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦}) | |
6 | 5 | eqeq2d 2748 | . . 3 ⊢ (𝑥 = 𝑦 → ({𝑥 ∣ 𝜑} = {𝑥} ↔ {𝑥 ∣ 𝜑} = {𝑦})) |
7 | 2, 4, 6 | cbvexv1 2339 | . 2 ⊢ (∃𝑥{𝑥 ∣ 𝜑} = {𝑥} ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦}) |
8 | 1, 7 | bitr4i 278 | 1 ⊢ (∃!𝑥𝜑 ↔ ∃𝑥{𝑥 ∣ 𝜑} = {𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1542 ∃wex 1782 ∃!weu 2567 {cab 2714 {csn 4591 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-nfc 2890 df-sn 4592 |
This theorem is referenced by: eusn 4696 uniintsn 4953 args 6049 opabiotadm 6928 mapsnd 8831 |
Copyright terms: Public domain | W3C validator |