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Mirrors > Home > MPE Home > Th. List > 0elon | Structured version Visualization version GIF version |
Description: The empty set is an ordinal number. Corollary 7N(b) of [Enderton] p. 193. (Contributed by NM, 17-Sep-1993.) |
Ref | Expression |
---|---|
0elon | ⊢ ∅ ∈ On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ord0 5694 | . 2 ⊢ Ord ∅ | |
2 | 0ex 4718 | . . 3 ⊢ ∅ ∈ V | |
3 | 2 | elon 5649 | . 2 ⊢ (∅ ∈ On ↔ Ord ∅) |
4 | 1, 3 | mpbir 220 | 1 ⊢ ∅ ∈ On |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 ∅c0 3874 Ord word 5639 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-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: inton 5699 onn0 5706 on0eqel 5762 orduninsuc 6935 onzsl 6938 smofvon2 7340 tfrlem16 7376 1on 7454 ordgt0ge1 7464 oa0 7483 om0 7484 oe0m 7485 oe0m0 7487 oe0 7489 oesuclem 7492 omcl 7503 oecl 7504 oa0r 7505 om0r 7506 oaord1 7518 oaword1 7519 oaword2 7520 oawordeu 7522 oa00 7526 odi 7546 oeoa 7564 oeoe 7566 nna0r 7576 nnm0r 7577 card2on 8342 card2inf 8343 harcl 8349 cantnfvalf 8445 rankon 8541 cardon 8653 card0 8667 alephon 8775 alephgeom 8788 alephfplem1 8810 cdafi 8895 ttukeylem4 9217 ttukeylem7 9220 cfpwsdom 9285 inar1 9476 rankcf 9478 gruina 9519 bnj168 30052 rdgprc0 30943 sltval2 31053 sltsolem1 31067 bdayelon 31079 rankeq1o 31448 0hf 31454 onsuccon 31607 onsucsuccmp 31613 finxp1o 32405 finxpreclem4 32407 harn0 36691 |
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