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Mirrors > Home > NFE Home > Th. List > nulel0c | GIF version |
Description: The empty set is a member of cardinal zero. (Contributed by SF, 13-Feb-2015.) |
Ref | Expression |
---|---|
nulel0c | ⊢ ∅ ∈ 0c |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2353 | . 2 ⊢ ∅ = ∅ | |
2 | el0c 4421 | . 2 ⊢ (∅ ∈ 0c ↔ ∅ = ∅) | |
3 | 1, 2 | mpbir 200 | 1 ⊢ ∅ ∈ 0c |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1642 ∈ wcel 1710 ∅c0 3550 0cc0c 4374 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-ss 3259 df-nul 3551 df-sn 3741 df-0c 4377 |
This theorem is referenced by: ncfinraise 4481 ncfinlower 4483 tfin0c 4497 nulnnn 4556 tc0c 6163 ce0nnul 6177 ce0nn 6180 ce0nulnc 6184 |
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