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Mirrors > Home > MPE Home > Th. List > ondomen | Structured version Visualization version GIF version |
Description: If a set is dominated by an ordinal, then it is numerable. (Contributed by Mario Carneiro, 5-Jan-2013.) |
Ref | Expression |
---|---|
ondomen | ⊢ ((𝐴 ∈ On ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ dom card) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 4933 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝐵 ≼ 𝑥 ↔ 𝐵 ≼ 𝐴)) | |
2 | 1 | rspcev 3535 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ≼ 𝐴) → ∃𝑥 ∈ On 𝐵 ≼ 𝑥) |
3 | ac10ct 9254 | . . 3 ⊢ (∃𝑥 ∈ On 𝐵 ≼ 𝑥 → ∃𝑟 𝑟 We 𝐵) | |
4 | 2, 3 | syl 17 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ≼ 𝐴) → ∃𝑟 𝑟 We 𝐵) |
5 | ween 9255 | . 2 ⊢ (𝐵 ∈ dom card ↔ ∃𝑟 𝑟 We 𝐵) | |
6 | 4, 5 | sylibr 226 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ dom card) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∃wex 1742 ∈ wcel 2050 ∃wrex 3089 class class class wbr 4929 We wwe 5365 dom cdm 5407 Oncon0 6029 ≼ cdom 8304 cardccrd 9158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-se 5367 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-isom 6197 df-riota 6937 df-wrecs 7750 df-recs 7812 df-en 8307 df-dom 8308 df-card 9162 |
This theorem is referenced by: numdom 9258 alephnbtwn2 9292 alephsucdom 9299 fictb 9465 cfslb2n 9488 gchaleph2 9892 hargch 9893 inawinalem 9909 rankcf 9997 tskuni 10003 1stcrestlem 21764 2ndcctbss 21767 2ndcomap 21770 2ndcsep 21771 tx1stc 21962 tx2ndc 21963 met2ndci 22835 |
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