![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > carddom2 | Structured version Visualization version GIF version |
Description: Two numerable sets have the dominance relationship iff their cardinalities have the subset relationship. See also carddom 9691, which uses AC. (Contributed by Mario Carneiro, 11-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.) |
Ref | Expression |
---|---|
carddom2 | ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ 𝐴 ≼ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | carddomi2 9109 | . 2 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴 ≼ 𝐵)) | |
2 | brdom2 8252 | . . 3 ⊢ (𝐴 ≼ 𝐵 ↔ (𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵)) | |
3 | cardon 9083 | . . . . . . . 8 ⊢ (card‘𝐴) ∈ On | |
4 | 3 | onelssi 6071 | . . . . . . 7 ⊢ ((card‘𝐵) ∈ (card‘𝐴) → (card‘𝐵) ⊆ (card‘𝐴)) |
5 | carddomi2 9109 | . . . . . . . 8 ⊢ ((𝐵 ∈ dom card ∧ 𝐴 ∈ dom card) → ((card‘𝐵) ⊆ (card‘𝐴) → 𝐵 ≼ 𝐴)) | |
6 | 5 | ancoms 452 | . . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐵) ⊆ (card‘𝐴) → 𝐵 ≼ 𝐴)) |
7 | domnsym 8355 | . . . . . . 7 ⊢ (𝐵 ≼ 𝐴 → ¬ 𝐴 ≺ 𝐵) | |
8 | 4, 6, 7 | syl56 36 | . . . . . 6 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐵) ∈ (card‘𝐴) → ¬ 𝐴 ≺ 𝐵)) |
9 | 8 | con2d 132 | . . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ≺ 𝐵 → ¬ (card‘𝐵) ∈ (card‘𝐴))) |
10 | cardon 9083 | . . . . . 6 ⊢ (card‘𝐵) ∈ On | |
11 | ontri1 5997 | . . . . . 6 ⊢ (((card‘𝐴) ∈ On ∧ (card‘𝐵) ∈ On) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ ¬ (card‘𝐵) ∈ (card‘𝐴))) | |
12 | 3, 10, 11 | mp2an 685 | . . . . 5 ⊢ ((card‘𝐴) ⊆ (card‘𝐵) ↔ ¬ (card‘𝐵) ∈ (card‘𝐴)) |
13 | 9, 12 | syl6ibr 244 | . . . 4 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ≺ 𝐵 → (card‘𝐴) ⊆ (card‘𝐵))) |
14 | carden2b 9106 | . . . . . 6 ⊢ (𝐴 ≈ 𝐵 → (card‘𝐴) = (card‘𝐵)) | |
15 | eqimss 3882 | . . . . . 6 ⊢ ((card‘𝐴) = (card‘𝐵) → (card‘𝐴) ⊆ (card‘𝐵)) | |
16 | 14, 15 | syl 17 | . . . . 5 ⊢ (𝐴 ≈ 𝐵 → (card‘𝐴) ⊆ (card‘𝐵)) |
17 | 16 | a1i 11 | . . . 4 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ≈ 𝐵 → (card‘𝐴) ⊆ (card‘𝐵))) |
18 | 13, 17 | jaod 892 | . . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵) → (card‘𝐴) ⊆ (card‘𝐵))) |
19 | 2, 18 | syl5bi 234 | . 2 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ≼ 𝐵 → (card‘𝐴) ⊆ (card‘𝐵))) |
20 | 1, 19 | impbid 204 | 1 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ 𝐴 ≼ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 ∨ wo 880 = wceq 1658 ∈ wcel 2166 ⊆ wss 3798 class class class wbr 4873 dom cdm 5342 Oncon0 5963 ‘cfv 6123 ≈ cen 8219 ≼ cdom 8220 ≺ csdm 8221 cardccrd 9074 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-ord 5966 df-on 5967 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-card 9078 |
This theorem is referenced by: carduni 9120 carden2 9126 cardsdom2 9127 domtri2 9128 infxpidm2 9153 cardaleph 9225 infenaleph 9227 alephinit 9231 ficardun2 9340 ackbij2 9380 cfflb 9396 fin1a2lem9 9545 carddom 9691 pwfseqlem5 9800 hashdom 13458 |
Copyright terms: Public domain | W3C validator |