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Mirrors > Home > MPE Home > Th. List > rmoeq | Structured version Visualization version GIF version |
Description: Equality's restricted existential "at most one" property. (Contributed by Thierry Arnoux, 30-Mar-2018.) (Revised by AV, 27-Oct-2020.) (Proof shortened by NM, 29-Oct-2020.) |
Ref | Expression |
---|---|
rmoeq | ⊢ ∃*𝑥 ∈ 𝐵 𝑥 = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | moeq 3642 | . . 3 ⊢ ∃*𝑥 𝑥 = 𝐴 | |
2 | 1 | moani 2553 | . 2 ⊢ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴) |
3 | df-rmo 3071 | . 2 ⊢ (∃*𝑥 ∈ 𝐵 𝑥 = 𝐴 ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝑥 = 𝐴)) | |
4 | 2, 3 | mpbir 230 | 1 ⊢ ∃*𝑥 ∈ 𝐵 𝑥 = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∃*wmo 2538 ∃*wrmo 3067 |
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 1913 ax-6 1971 ax-7 2011 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1783 df-mo 2540 df-cleq 2730 df-rmo 3071 |
This theorem is referenced by: nbusgredgeu 27733 |
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