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Mirrors > Home > MPE Home > Th. List > Mathboxes > euabsneu | Structured version Visualization version GIF version |
Description: Another way to express existential uniqueness of a wff 𝜑: its associated class abstraction {𝑥 ∣ 𝜑} is a singleton. Variant of euabsn2 4661 using existential uniqueness for the singleton element instead of existence only. (Contributed by AV, 24-Aug-2022.) |
Ref | Expression |
---|---|
euabsneu | ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦{𝑥 ∣ 𝜑} = {𝑦}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mosneq 4773 | . . . 4 ⊢ ∃*𝑦{𝑦} = {𝑥 ∣ 𝜑} | |
2 | eqcom 2828 | . . . . 5 ⊢ ({𝑦} = {𝑥 ∣ 𝜑} ↔ {𝑥 ∣ 𝜑} = {𝑦}) | |
3 | 2 | mobii 2631 | . . . 4 ⊢ (∃*𝑦{𝑦} = {𝑥 ∣ 𝜑} ↔ ∃*𝑦{𝑥 ∣ 𝜑} = {𝑦}) |
4 | 1, 3 | mpbi 232 | . . 3 ⊢ ∃*𝑦{𝑥 ∣ 𝜑} = {𝑦} |
5 | 4 | biantru 532 | . 2 ⊢ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} ↔ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} ∧ ∃*𝑦{𝑥 ∣ 𝜑} = {𝑦})) |
6 | euabsn2 4661 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦{𝑥 ∣ 𝜑} = {𝑦}) | |
7 | df-eu 2654 | . 2 ⊢ (∃!𝑦{𝑥 ∣ 𝜑} = {𝑦} ↔ (∃𝑦{𝑥 ∣ 𝜑} = {𝑦} ∧ ∃*𝑦{𝑥 ∣ 𝜑} = {𝑦})) | |
8 | 5, 6, 7 | 3bitr4i 305 | 1 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦{𝑥 ∣ 𝜑} = {𝑦}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∃wex 1780 ∃*wmo 2620 ∃!weu 2653 {cab 2799 {csn 4567 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-sn 4568 |
This theorem is referenced by: reuaiotaiota 43337 |
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