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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 4724 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦}) | |
2 | nfv 1909 | . . 3 ⊢ Ⅎ𝑦{𝑥 ∣ 𝜑} = {𝑥} | |
3 | nfab1 2899 | . . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | |
4 | 3 | nfeq1 2912 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} = {𝑦} |
5 | sneq 4633 | . . . 4 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦}) | |
6 | 5 | eqeq2d 2737 | . . 3 ⊢ (𝑥 = 𝑦 → ({𝑥 ∣ 𝜑} = {𝑥} ↔ {𝑥 ∣ 𝜑} = {𝑦})) |
7 | 2, 4, 6 | cbvexv1 2332 | . 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 1533 ∃wex 1773 ∃!weu 2556 {cab 2703 {csn 4623 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-nfc 2879 df-sn 4624 |
This theorem is referenced by: eusn 4729 uniintsn 4984 args 6084 opabiotadm 6966 mapsnd 8879 |
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