|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > en1eqsnbi | Structured version Visualization version GIF version | ||
| Description: A set containing an element has exactly one element iff it is a singleton. Formerly part of proof for rngen1zr 20778. (Contributed by FL, 13-Feb-2010.) (Revised by AV, 25-Jan-2020.) | 
| Ref | Expression | 
|---|---|
| en1eqsnbi | ⊢ (𝐴 ∈ 𝐵 → (𝐵 ≈ 1o ↔ 𝐵 = {𝐴})) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | en1eqsn 9308 | . . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → 𝐵 = {𝐴}) | |
| 2 | 1 | ex 412 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐵 ≈ 1o → 𝐵 = {𝐴})) | 
| 3 | ensn1g 9062 | . . 3 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ≈ 1o) | |
| 4 | breq1 5146 | . . 3 ⊢ (𝐵 = {𝐴} → (𝐵 ≈ 1o ↔ {𝐴} ≈ 1o)) | |
| 5 | 3, 4 | syl5ibrcom 247 | . 2 ⊢ (𝐴 ∈ 𝐵 → (𝐵 = {𝐴} → 𝐵 ≈ 1o)) | 
| 6 | 2, 5 | impbid 212 | 1 ⊢ (𝐴 ∈ 𝐵 → (𝐵 ≈ 1o ↔ 𝐵 = {𝐴})) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 {csn 4626 class class class wbr 5143 1oc1o 8499 ≈ cen 8982 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-1o 8506 df-en 8986 | 
| This theorem is referenced by: srgen1zr 20213 rngen1zr 20778 rngosn4 37932 | 
| Copyright terms: Public domain | W3C validator |