|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > euen1 | Structured version Visualization version GIF version | ||
| Description: Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.) | 
| Ref | Expression | 
|---|---|
| euen1 | ⊢ (∃!𝑥𝜑 ↔ {𝑥 ∣ 𝜑} ≈ 1o) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | reuen1 9067 | . 2 ⊢ (∃!𝑥 ∈ V 𝜑 ↔ {𝑥 ∈ V ∣ 𝜑} ≈ 1o) | |
| 2 | reuv 3509 | . 2 ⊢ (∃!𝑥 ∈ V 𝜑 ↔ ∃!𝑥𝜑) | |
| 3 | rabab 3511 | . . 3 ⊢ {𝑥 ∈ V ∣ 𝜑} = {𝑥 ∣ 𝜑} | |
| 4 | 3 | breq1i 5149 | . 2 ⊢ ({𝑥 ∈ V ∣ 𝜑} ≈ 1o ↔ {𝑥 ∣ 𝜑} ≈ 1o) | 
| 5 | 1, 2, 4 | 3bitr3i 301 | 1 ⊢ (∃!𝑥𝜑 ↔ {𝑥 ∣ 𝜑} ≈ 1o) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∃!weu 2567 {cab 2713 ∃!wreu 3377 {crab 3435 Vcvv 3479 class class class wbr 5142 1oc1o 8500 ≈ cen 8983 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-1o 8507 df-en 8987 | 
| This theorem is referenced by: euen1b 9069 modom 9281 | 
| Copyright terms: Public domain | W3C validator |