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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 5152 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝐵 ≼ 𝑥 ↔ 𝐵 ≼ 𝐴)) | |
2 | 1 | rspcev 3612 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ≼ 𝐴) → ∃𝑥 ∈ On 𝐵 ≼ 𝑥) |
3 | ac10ct 10035 | . . 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 10036 | . 2 ⊢ (𝐵 ∈ dom card ↔ ∃𝑟 𝑟 We 𝐵) | |
6 | 4, 5 | sylibr 233 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ dom card) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∃wex 1780 ∈ wcel 2105 ∃wrex 3069 class class class wbr 5148 We wwe 5630 dom cdm 5676 Oncon0 6364 ≼ cdom 8943 cardccrd 9936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-en 8946 df-dom 8947 df-card 9940 |
This theorem is referenced by: numdom 10039 alephnbtwn2 10073 alephsucdom 10080 fictb 10246 cfslb2n 10269 gchaleph2 10673 hargch 10674 inawinalem 10690 rankcf 10778 tskuni 10784 1stcrestlem 23276 2ndcctbss 23279 2ndcomap 23282 2ndcsep 23283 tx1stc 23474 tx2ndc 23475 met2ndci 24351 rn1st 44439 |
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