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Mirrors > Home > MPE Home > Th. List > euen1b | Structured version Visualization version GIF version |
Description: Two ways to express "𝐴 has a unique element". (Contributed by Mario Carneiro, 9-Apr-2015.) |
Ref | Expression |
---|---|
euen1b | ⊢ (𝐴 ≈ 1𝑜 ↔ ∃!𝑥 𝑥 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | euen1 8067 | . 2 ⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ {𝑥 ∣ 𝑥 ∈ 𝐴} ≈ 1𝑜) | |
2 | abid2 2774 | . . 3 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 | |
3 | 2 | breq1i 4692 | . 2 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} ≈ 1𝑜 ↔ 𝐴 ≈ 1𝑜) |
4 | 1, 3 | bitr2i 265 | 1 ⊢ (𝐴 ≈ 1𝑜 ↔ ∃!𝑥 𝑥 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∈ wcel 2030 ∃!weu 2498 {cab 2637 class class class wbr 4685 1𝑜c1o 7598 ≈ cen 7994 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-1o 7605 df-en 7998 |
This theorem is referenced by: euhash1 13246 f1otrspeq 17913 hausflf2 21849 minveclem4a 23247 |
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