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Mirrors > Home > MPE Home > Th. List > 0elon | Structured version Visualization version Unicode 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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ord0 5777 | . 2 | |
2 | 0ex 4790 | . . 3 | |
3 | 2 | elon 5732 | . 2 |
4 | 1, 3 | mpbir 221 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wcel 1990 c0 3915 word 5722 con0 5723 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-in 3581 df-ss 3588 df-nul 3916 df-pw 4160 df-uni 4437 df-tr 4753 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-ord 5726 df-on 5727 |
This theorem is referenced by: inton 5782 onn0 5789 on0eqel 5845 orduninsuc 7043 onzsl 7046 smofvon2 7453 tfrlem16 7489 1on 7567 ordgt0ge1 7577 oa0 7596 om0 7597 oe0m 7598 oe0m0 7600 oe0 7602 oesuclem 7605 omcl 7616 oecl 7617 oa0r 7618 om0r 7619 oaord1 7631 oaword1 7632 oaword2 7633 oawordeu 7635 oa00 7639 odi 7659 oeoa 7677 oeoe 7679 nna0r 7689 nnm0r 7690 card2on 8459 card2inf 8460 harcl 8466 cantnfvalf 8562 rankon 8658 cardon 8770 card0 8784 alephon 8892 alephgeom 8905 alephfplem1 8927 cdafi 9012 ttukeylem4 9334 ttukeylem7 9337 cfpwsdom 9406 inar1 9597 rankcf 9599 gruina 9640 bnj168 30798 rdgprc0 31699 sltval2 31809 sltsolem1 31826 nosepnelem 31830 nodense 31842 nolt02o 31845 bdayelon 31892 rankeq1o 32278 0hf 32284 onsucconn 32437 onsucsuccmp 32443 finxp1o 33229 finxpreclem4 33231 harn0 37672 |
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