![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
Mirrors > Home > MPE Home > Th. List > 0pss | Structured version Unicode version |
Description: The null set is a proper subset of any nonempty set. (Contributed by NM, 27-Feb-1996.) |
Ref | Expression |
---|---|
0pss |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 3777 |
. . 3
![]() ![]() ![]() ![]() | |
2 | df-pss 3455 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
3 | 1, 2 | mpbiran 909 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4 | necom 2721 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | 3, 4 | bitri 249 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 |
This theorem depends on definitions: df-bi 185 df-an 371 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2650 df-v 3080 df-dif 3442 df-in 3446 df-ss 3453 df-pss 3455 df-nul 3749 |
This theorem is referenced by: php 7608 zornn0g 8788 prn0 9272 genpn0 9286 nqpr 9297 ltexprlem5 9323 reclem2pr 9331 suplem1pr 9335 alexsubALTlem4 19757 bj-2upln0 32868 bj-2upln1upl 32869 |
Copyright terms: Public domain | W3C validator |