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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 5062 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝐵 ≼ 𝑥 ↔ 𝐵 ≼ 𝐴)) | |
2 | 1 | rspcev 3622 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ≼ 𝐴) → ∃𝑥 ∈ On 𝐵 ≼ 𝑥) |
3 | ac10ct 9454 | . . 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 9455 | . 2 ⊢ (𝐵 ∈ dom card ↔ ∃𝑟 𝑟 We 𝐵) | |
6 | 4, 5 | sylibr 236 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ dom card) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∃wex 1776 ∈ wcel 2110 ∃wrex 3139 class class class wbr 5058 We wwe 5507 dom cdm 5549 Oncon0 6185 ≼ cdom 8501 cardccrd 9358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-wrecs 7941 df-recs 8002 df-en 8504 df-dom 8505 df-card 9362 |
This theorem is referenced by: numdom 9458 alephnbtwn2 9492 alephsucdom 9499 fictb 9661 cfslb2n 9684 gchaleph2 10088 hargch 10089 inawinalem 10105 rankcf 10193 tskuni 10199 1stcrestlem 22054 2ndcctbss 22057 2ndcomap 22060 2ndcsep 22061 tx1stc 22252 tx2ndc 22253 met2ndci 23126 |
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