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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 5061 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝐵 ≼ 𝑥 ↔ 𝐵 ≼ 𝐴)) | |
2 | 1 | rspcev 3620 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ≼ 𝐴) → ∃𝑥 ∈ On 𝐵 ≼ 𝑥) |
3 | ac10ct 9448 | . . 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 9449 | . 2 ⊢ (𝐵 ∈ dom card ↔ ∃𝑟 𝑟 We 𝐵) | |
6 | 4, 5 | sylibr 235 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ dom card) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∃wex 1771 ∈ wcel 2105 ∃wrex 3136 class class class wbr 5057 We wwe 5506 dom cdm 5548 Oncon0 6184 ≼ cdom 8495 cardccrd 9352 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-wrecs 7936 df-recs 7997 df-en 8498 df-dom 8499 df-card 9356 |
This theorem is referenced by: numdom 9452 alephnbtwn2 9486 alephsucdom 9493 fictb 9655 cfslb2n 9678 gchaleph2 10082 hargch 10083 inawinalem 10099 rankcf 10187 tskuni 10193 1stcrestlem 21988 2ndcctbss 21991 2ndcomap 21994 2ndcsep 21995 tx1stc 22186 tx2ndc 22187 met2ndci 23059 |
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