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Mirrors > Home > MPE Home > Th. List > harsdom | Structured version Visualization version GIF version |
Description: The Hartogs number of a well-orderable set strictly dominates the set. (Contributed by Mario Carneiro, 15-May-2015.) |
Ref | Expression |
---|---|
harsdom | ⊢ (𝐴 ∈ dom card → 𝐴 ≺ (har‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | harndom 8760 | . 2 ⊢ ¬ (har‘𝐴) ≼ 𝐴 | |
2 | harcl 8757 | . . . 4 ⊢ (har‘𝐴) ∈ On | |
3 | onenon 9110 | . . . 4 ⊢ ((har‘𝐴) ∈ On → (har‘𝐴) ∈ dom card) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ (har‘𝐴) ∈ dom card |
5 | domtri2 9150 | . . . 4 ⊢ (((har‘𝐴) ∈ dom card ∧ 𝐴 ∈ dom card) → ((har‘𝐴) ≼ 𝐴 ↔ ¬ 𝐴 ≺ (har‘𝐴))) | |
6 | 5 | con2bid 346 | . . 3 ⊢ (((har‘𝐴) ∈ dom card ∧ 𝐴 ∈ dom card) → (𝐴 ≺ (har‘𝐴) ↔ ¬ (har‘𝐴) ≼ 𝐴)) |
7 | 4, 6 | mpan 680 | . 2 ⊢ (𝐴 ∈ dom card → (𝐴 ≺ (har‘𝐴) ↔ ¬ (har‘𝐴) ≼ 𝐴)) |
8 | 1, 7 | mpbiri 250 | 1 ⊢ (𝐴 ∈ dom card → 𝐴 ≺ (har‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 ∈ wcel 2107 class class class wbr 4888 dom cdm 5357 Oncon0 5978 ‘cfv 6137 ≼ cdom 8241 ≺ csdm 8242 harchar 8752 cardccrd 9096 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-wrecs 7691 df-recs 7753 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-oi 8706 df-har 8754 df-card 9100 |
This theorem is referenced by: onsdom 9157 harval2 9158 alephordilem1 9231 gchaleph2 9831 |
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