Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > inton | Structured version Visualization version GIF version |
Description: The intersection of the class of ordinal numbers is the empty set. (Contributed by NM, 20-Oct-2003.) |
Ref | Expression |
---|---|
inton | ⊢ ∩ On = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0elon 6237 | . 2 ⊢ ∅ ∈ On | |
2 | int0el 4898 | . 2 ⊢ (∅ ∈ On → ∩ On = ∅) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ∩ On = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 ∅c0 4288 ∩ cint 4867 Oncon0 6184 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-nul 5201 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-in 3940 df-ss 3949 df-nul 4289 df-pw 4537 df-uni 4831 df-int 4868 df-tr 5164 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-ord 6187 df-on 6188 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |