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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 3512 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | cardeqv 9891 | . 2 ⊢ dom card = V | |
3 | 1, 2 | eleqtrrdi 2924 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ dom card) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 Vcvv 3494 dom cdm 5555 cardccrd 9364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-ac2 9885 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-wrecs 7947 df-recs 8008 df-en 8510 df-card 9368 df-ac 9542 |
This theorem is referenced by: numth2 9893 ac5b 9900 ac6 9902 zorn2 9928 zorn 9929 zornn0 9930 ttukey 9940 fodom 9944 wdomac 9949 iundom 9964 cardval 9968 cardid 9969 carden 9973 carddom 9976 cardsdom 9977 domtri 9978 sdomsdomcard 9982 infxpidm 9984 ondomon 9985 infmap 9998 aleph1irr 15599 lbsext 19935 hauspwdom 22109 filssufil 22520 ufilen 22538 |
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