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Mirrors > Home > MPE Home > Th. List > onsdom | Structured version Visualization version GIF version |
Description: Any well-orderable set is strictly dominated by an ordinal number. (Contributed by Jeff Hankins, 22-Oct-2009.) (Proof shortened by Mario Carneiro, 15-May-2015.) |
Ref | Expression |
---|---|
onsdom | ⊢ (𝐴 ∈ dom card → ∃𝑥 ∈ On 𝐴 ≺ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | harcl 9560 | . 2 ⊢ (har‘𝐴) ∈ On | |
2 | harsdom 9996 | . 2 ⊢ (𝐴 ∈ dom card → 𝐴 ≺ (har‘𝐴)) | |
3 | breq2 5152 | . . 3 ⊢ (𝑥 = (har‘𝐴) → (𝐴 ≺ 𝑥 ↔ 𝐴 ≺ (har‘𝐴))) | |
4 | 3 | rspcev 3612 | . 2 ⊢ (((har‘𝐴) ∈ On ∧ 𝐴 ≺ (har‘𝐴)) → ∃𝑥 ∈ On 𝐴 ≺ 𝑥) |
5 | 1, 2, 4 | sylancr 586 | 1 ⊢ (𝐴 ∈ dom card → ∃𝑥 ∈ On 𝐴 ≺ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ∃wrex 3069 class class class wbr 5148 dom cdm 5676 Oncon0 6364 ‘cfv 6543 ≺ csdm 8944 harchar 9557 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-lim 6369 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-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-oi 9511 df-har 9558 df-card 9940 |
This theorem is referenced by: (None) |
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