![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > sdomnen | Structured version Visualization version GIF version |
Description: Strict dominance implies non-equinumerosity. (Contributed by NM, 10-Jun-1998.) |
Ref | Expression |
---|---|
sdomnen | ⊢ (𝐴 ≺ 𝐵 → ¬ 𝐴 ≈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brsdom 9014 | . 2 ⊢ (𝐴 ≺ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≈ 𝐵)) | |
2 | 1 | simprbi 496 | 1 ⊢ (𝐴 ≺ 𝐵 → ¬ 𝐴 ≈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 class class class wbr 5148 ≈ cen 8981 ≼ cdom 8982 ≺ csdm 8983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-dif 3966 df-br 5149 df-sdom 8987 |
This theorem is referenced by: bren2 9022 domdifsn 9093 sucdom2OLD 9121 sdomnsym 9137 domnsym 9138 sdomirr 9153 domnsymfi 9238 sucdom2 9241 php5 9249 phpeqd 9250 phpeqdOLD 9260 1sdom2dom 9281 pssinf 9290 f1finf1o 9303 f1finf1oOLD 9304 isfinite2 9332 cardom 10024 pm54.43 10039 pr2neOLD 10043 alephdom 10119 cdainflem 10226 ackbij1b 10276 isfin4p1 10353 fin23lem25 10362 fin67 10433 axcclem 10495 canthp1lem2 10691 gchinf 10695 pwfseqlem4 10700 tskssel 10795 1nprm 16713 en2top 23008 domalom 37387 pibt2 37400 rp-isfinite6 43508 ensucne0OLD 43520 iscard5 43526 omiscard 43533 |
Copyright terms: Public domain | W3C validator |