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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 6246 | . 2 ⊢ ∅ ∈ On | |
2 | int0el 4909 | . 2 ⊢ (∅ ∈ On → ∩ On = ∅) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ∩ On = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 ∅c0 4293 ∩ cint 4878 Oncon0 6193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-nul 5212 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-in 3945 df-ss 3954 df-nul 4294 df-pw 4543 df-uni 4841 df-int 4879 df-tr 5175 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-ord 6196 df-on 6197 |
This theorem is referenced by: (None) |
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