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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 5695 | . 2 ⊢ ∅ ∈ On | |
2 | int0el 4443 | . 2 ⊢ (∅ ∈ On → ∩ On = ∅) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ∩ On = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 ∅c0 3874 ∩ cint 4410 Oncon0 5640 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-in 3547 df-ss 3554 df-nul 3875 df-pw 4110 df-uni 4373 df-int 4411 df-tr 4681 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-ord 5643 df-on 5644 |
This theorem is referenced by: (None) |
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