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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 4204 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦}) | |
2 | nfv 1830 | . . 3 ⊢ Ⅎ𝑦{𝑥 ∣ 𝜑} = {𝑥} | |
3 | nfab1 2753 | . . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} | |
4 | 3 | nfeq1 2764 | . . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑} = {𝑦} |
5 | sneq 4135 | . . . 4 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦}) | |
6 | 5 | eqeq2d 2620 | . . 3 ⊢ (𝑥 = 𝑦 → ({𝑥 ∣ 𝜑} = {𝑥} ↔ {𝑥 ∣ 𝜑} = {𝑦})) |
7 | 2, 4, 6 | cbvex 2260 | . 2 ⊢ (∃𝑥{𝑥 ∣ 𝜑} = {𝑥} ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦}) |
8 | 1, 7 | bitr4i 266 | 1 ⊢ (∃!𝑥𝜑 ↔ ∃𝑥{𝑥 ∣ 𝜑} = {𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∃wex 1695 ∃!weu 2458 {cab 2596 {csn 4125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-sn 4126 |
This theorem is referenced by: eusn 4209 uniintsn 4449 args 5412 opabiotadm 6170 mapsn 7785 mapsnd 38383 |
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