![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > orddisj | Structured version Visualization version GIF version |
Description: An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998.) |
Ref | Expression |
---|---|
orddisj | ⊢ (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordirr 6383 | . 2 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴) | |
2 | disjsn 4716 | . 2 ⊢ ((𝐴 ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ 𝐴) | |
3 | 1, 2 | sylibr 233 | 1 ⊢ (Ord 𝐴 → (𝐴 ∩ {𝐴}) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1542 ∈ wcel 2107 ∩ cin 3948 ∅c0 4323 {csn 4629 Ord word 6364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-eprel 5581 df-fr 5632 df-we 5634 df-ord 6368 |
This theorem is referenced by: orddif 6461 omsucne 7874 tfrlem10 8387 enrefnn 9047 pssnn 9168 unfi 9172 phplem2OLD 9218 isinf 9260 isinfOLD 9261 pssnnOLD 9265 dif1ennnALT 9277 ackbij1lem5 10219 ackbij1lem14 10228 ackbij1lem16 10230 unsnen 10548 pwfi2f1o 41838 |
Copyright terms: Public domain | W3C validator |