Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > numth3 | Structured version Visualization version GIF version |
Description: All sets are well-orderable under choice. (Contributed by Stefan O'Rear, 28-Feb-2015.) |
Ref | Expression |
---|---|
numth3 | ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ dom card) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3510 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | cardeqv 9879 | . 2 ⊢ dom card = V | |
3 | 1, 2 | eleqtrrdi 2921 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ dom card) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 Vcvv 3492 dom cdm 5548 cardccrd 9352 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-ac2 9873 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-wrecs 7936 df-recs 7997 df-en 8498 df-card 9356 df-ac 9530 |
This theorem is referenced by: numth2 9881 ac5b 9888 ac6 9890 zorn2 9916 zorn 9917 zornn0 9918 ttukey 9928 fodom 9932 wdomac 9937 iundom 9952 cardval 9956 cardid 9957 carden 9961 carddom 9964 cardsdom 9965 domtri 9966 sdomsdomcard 9970 infxpidm 9972 ondomon 9973 infmap 9986 aleph1irr 15587 lbsext 19864 hauspwdom 22037 filssufil 22448 ufilen 22466 |
Copyright terms: Public domain | W3C validator |