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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 3414 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | cardeqv 9626 | . 2 ⊢ dom card = V | |
3 | 1, 2 | syl6eleqr 2870 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ dom card) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 Vcvv 3398 dom cdm 5355 cardccrd 9094 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-ac2 9620 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-wrecs 7689 df-recs 7751 df-en 8242 df-card 9098 df-ac 9272 |
This theorem is referenced by: numth2 9628 ac5b 9635 ac6 9637 zorn2 9663 zorn 9664 zornn0 9665 ttukey 9675 fodom 9679 wdomac 9684 iundom 9699 cardval 9703 cardid 9704 carden 9708 carddom 9711 cardsdom 9712 domtri 9713 sdomsdomcard 9717 infxpidm 9719 ondomon 9720 infmap 9733 aleph1irr 15379 lbsext 19560 hauspwdom 21713 filssufil 22124 ufilen 22142 |
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