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Mirrors > Home > MPE Home > Th. List > infcntss | Structured version Visualization version Unicode version |
Description: Every infinite set has a denumerable subset. Similar to Exercise 8 of [TakeutiZaring] p. 91. (However, we need neither AC nor the Axiom of Infinity because of the way we express "infinite" in the antecedent.) (Contributed by NM, 23-Oct-2004.) |
Ref | Expression |
---|---|
infcntss.1 |
Ref | Expression |
---|---|
infcntss |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | infcntss.1 | . . 3 | |
2 | 1 | domen 7968 | . 2 |
3 | ensym 8005 | . . . . 5 | |
4 | 3 | anim2i 593 | . . . 4 |
5 | 4 | ancoms 469 | . . 3 |
6 | 5 | eximi 1762 | . 2 |
7 | 2, 6 | sylbi 207 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wex 1704 wcel 1990 cvv 3200 wss 3574 class class class wbr 4653 com 7065 cen 7952 cdom 7953 |
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-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-er 7742 df-en 7956 df-dom 7957 |
This theorem is referenced by: (None) |
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