| Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > ILE Home > Th. List > rnsnm | GIF version | ||
| Description: The range of a singleton is inhabited iff the singleton argument is an ordered pair. (Contributed by Jim Kingdon, 15-Dec-2018.) |
| Ref | Expression |
|---|---|
| rnsnm | ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥 𝑥 ∈ ran {𝐴}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dmsnm 5148 | . 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥 𝑥 ∈ dom {𝐴}) | |
| 2 | dmmrnm 4897 | . 2 ⊢ (∃𝑥 𝑥 ∈ dom {𝐴} ↔ ∃𝑥 𝑥 ∈ ran {𝐴}) | |
| 3 | 1, 2 | bitri 184 | 1 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥 𝑥 ∈ ran {𝐴}) |
| Colors of variables: wff set class |
| Syntax hints: ↔ wb 105 ∃wex 1515 ∈ wcel 2176 Vcvv 2772 {csn 3633 × cxp 4673 dom cdm 4675 ran crn 4676 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-14 2179 ax-ext 2187 ax-sep 4162 ax-pow 4218 ax-pr 4253 |
| This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-v 2774 df-un 3170 df-in 3172 df-ss 3179 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-br 4045 df-opab 4106 df-xp 4681 df-cnv 4683 df-dm 4685 df-rn 4686 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |