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Theorem carddom2 10015
Description: Two numerable sets have the dominance relationship iff their cardinalities have the subset relationship. See also carddom 10592, which uses AC. (Contributed by Mario Carneiro, 11-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
carddom2 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ 𝐴𝐵))

Proof of Theorem carddom2
StepHypRef Expression
1 carddomi2 10008 . 2 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴𝐵))
2 brdom2 9021 . . 3 (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵))
3 cardon 9982 . . . . . . . 8 (card‘𝐴) ∈ On
43onelssi 6501 . . . . . . 7 ((card‘𝐵) ∈ (card‘𝐴) → (card‘𝐵) ⊆ (card‘𝐴))
5 carddomi2 10008 . . . . . . . 8 ((𝐵 ∈ dom card ∧ 𝐴 ∈ dom card) → ((card‘𝐵) ⊆ (card‘𝐴) → 𝐵𝐴))
65ancoms 458 . . . . . . 7 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐵) ⊆ (card‘𝐴) → 𝐵𝐴))
7 domnsym 9138 . . . . . . 7 (𝐵𝐴 → ¬ 𝐴𝐵)
84, 6, 7syl56 36 . . . . . 6 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐵) ∈ (card‘𝐴) → ¬ 𝐴𝐵))
98con2d 134 . . . . 5 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵 → ¬ (card‘𝐵) ∈ (card‘𝐴)))
10 cardon 9982 . . . . . 6 (card‘𝐵) ∈ On
11 ontri1 6420 . . . . . 6 (((card‘𝐴) ∈ On ∧ (card‘𝐵) ∈ On) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ ¬ (card‘𝐵) ∈ (card‘𝐴)))
123, 10, 11mp2an 692 . . . . 5 ((card‘𝐴) ⊆ (card‘𝐵) ↔ ¬ (card‘𝐵) ∈ (card‘𝐴))
139, 12imbitrrdi 252 . . . 4 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵 → (card‘𝐴) ⊆ (card‘𝐵)))
14 carden2b 10005 . . . . . 6 (𝐴𝐵 → (card‘𝐴) = (card‘𝐵))
15 eqimss 4054 . . . . . 6 ((card‘𝐴) = (card‘𝐵) → (card‘𝐴) ⊆ (card‘𝐵))
1614, 15syl 17 . . . . 5 (𝐴𝐵 → (card‘𝐴) ⊆ (card‘𝐵))
1716a1i 11 . . . 4 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵 → (card‘𝐴) ⊆ (card‘𝐵)))
1813, 17jaod 859 . . 3 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((𝐴𝐵𝐴𝐵) → (card‘𝐴) ⊆ (card‘𝐵)))
192, 18biimtrid 242 . 2 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵 → (card‘𝐴) ⊆ (card‘𝐵)))
201, 19impbid 212 1 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847   = wceq 1537  wcel 2106  wss 3963   class class class wbr 5148  dom cdm 5689  Oncon0 6386  cfv 6563  cen 8981  cdom 8982  csdm 8983  cardccrd 9973
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-card 9977
This theorem is referenced by:  carduni  10019  carden2  10025  cardsdom2  10026  domtri2  10027  infxpidm2  10055  cardaleph  10127  infenaleph  10129  alephinit  10133  ficardun2  10240  ackbij2  10280  cfflb  10297  fin1a2lem9  10446  carddom  10592  pwfseqlem5  10701  hashdom  14415  minregex2  43525
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