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Mirrors > Home > NFE Home > Th. List > el0c | GIF version |
Description: Membership in cardinal zero. (Contributed by SF, 22-Jan-2015.) |
Ref | Expression |
---|---|
el0c | ⊢ (A ∈ 0c ↔ A = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-0c 4377 | . . 3 ⊢ 0c = {∅} | |
2 | 1 | eleq2i 2417 | . 2 ⊢ (A ∈ 0c ↔ A ∈ {∅}) |
3 | 0ex 4110 | . . 3 ⊢ ∅ ∈ V | |
4 | 3 | elsnc2 3762 | . 2 ⊢ (A ∈ {∅} ↔ A = ∅) |
5 | 2, 4 | bitri 240 | 1 ⊢ (A ∈ 0c ↔ A = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 = wceq 1642 ∈ wcel 1710 ∅c0 3550 {csn 3737 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: nulel0c 4422 0fin 4423 ncfinraise 4481 ncfinlower 4483 nnadjoin 4520 nnpweq 4523 sfin01 4528 tfinnn 4534 sfinltfin 4535 vfin1cltv 4547 df0c2 6137 ncvsq 6256 0lt1c 6258 nmembers1lem2 6269 |
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