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Mirrors > Home > ILE Home > Th. List > cardonle | Unicode version |
Description: The cardinal of an ordinal number is less than or equal to the ordinal number. Proposition 10.6(3) of [TakeutiZaring] p. 85. (Contributed by NM, 22-Oct-2003.) |
Ref | Expression |
---|---|
cardonle |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oncardval 7133 | . 2 | |
2 | enrefg 6721 | . . 3 | |
3 | breq1 3979 | . . . 4 | |
4 | 3 | intminss 3843 | . . 3 |
5 | 2, 4 | mpdan 418 | . 2 |
6 | 1, 5 | eqsstrd 3173 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wcel 2135 crab 2446 wss 3111 cint 3818 class class class wbr 3976 con0 4335 cfv 5182 cen 6695 ccrd 7126 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-en 6698 df-card 7127 |
This theorem is referenced by: card0 7135 |
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