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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 9970, 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 9393 | . 2 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴 ≼ 𝐵)) | |
2 | brdom2 8533 | . . 3 ⊢ (𝐴 ≼ 𝐵 ↔ (𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵)) | |
3 | cardon 9367 | . . . . . . . 8 ⊢ (card‘𝐴) ∈ On | |
4 | 3 | onelssi 6293 | . . . . . . 7 ⊢ ((card‘𝐵) ∈ (card‘𝐴) → (card‘𝐵) ⊆ (card‘𝐴)) |
5 | carddomi2 9393 | . . . . . . . 8 ⊢ ((𝐵 ∈ dom card ∧ 𝐴 ∈ dom card) → ((card‘𝐵) ⊆ (card‘𝐴) → 𝐵 ≼ 𝐴)) | |
6 | 5 | ancoms 461 | . . . . . . 7 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐵) ⊆ (card‘𝐴) → 𝐵 ≼ 𝐴)) |
7 | domnsym 8637 | . . . . . . 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 136 | . . . . 5 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ≺ 𝐵 → ¬ (card‘𝐵) ∈ (card‘𝐴))) |
10 | cardon 9367 | . . . . . 6 ⊢ (card‘𝐵) ∈ On | |
11 | ontri1 6219 | . . . . . 6 ⊢ (((card‘𝐴) ∈ On ∧ (card‘𝐵) ∈ On) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ ¬ (card‘𝐵) ∈ (card‘𝐴))) | |
12 | 3, 10, 11 | mp2an 690 | . . . . 5 ⊢ ((card‘𝐴) ⊆ (card‘𝐵) ↔ ¬ (card‘𝐵) ∈ (card‘𝐴)) |
13 | 9, 12 | syl6ibr 254 | . . . 4 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ≺ 𝐵 → (card‘𝐴) ⊆ (card‘𝐵))) |
14 | carden2b 9390 | . . . . . 6 ⊢ (𝐴 ≈ 𝐵 → (card‘𝐴) = (card‘𝐵)) | |
15 | eqimss 4022 | . . . . . 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 855 | . . 3 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵) → (card‘𝐴) ⊆ (card‘𝐵))) |
19 | 2, 18 | syl5bi 244 | . 2 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 ≼ 𝐵 → (card‘𝐴) ⊆ (card‘𝐵))) |
20 | 1, 19 | impbid 214 | 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 208 ∧ wa 398 ∨ wo 843 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 class class class wbr 5058 dom cdm 5549 Oncon0 6185 ‘cfv 6349 ≈ cen 8500 ≼ cdom 8501 ≺ csdm 8502 cardccrd 9358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-ord 6188 df-on 6189 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-card 9362 |
This theorem is referenced by: carduni 9404 carden2 9410 cardsdom2 9411 domtri2 9412 infxpidm2 9437 cardaleph 9509 infenaleph 9511 alephinit 9515 ficardun2 9619 ackbij2 9659 cfflb 9675 fin1a2lem9 9824 carddom 9970 pwfseqlem5 10079 hashdom 13734 |
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