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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 3185 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | cardeqv 9174 | . 2 ⊢ dom card = V | |
3 | 1, 2 | syl6eleqr 2699 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ dom card) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 Vcvv 3173 dom cdm 5038 cardccrd 8644 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-ac2 9168 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-wrecs 7294 df-recs 7355 df-en 7842 df-card 8648 df-ac 8822 |
This theorem is referenced by: numth2 9176 ac5b 9183 ac6 9185 zorn2 9211 zorn 9212 zornn0 9213 ttukey 9223 fodom 9227 wdomac 9230 iundom 9243 cardval 9247 cardid 9248 carden 9252 carddom 9255 cardsdom 9256 domtri 9257 sdomsdomcard 9261 infxpidm 9263 ondomon 9264 infmap 9277 aleph1irr 14814 lbsext 18984 hauspwdom 21114 filssufil 21526 ufilen 21544 |
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