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Theorem carddom2 8663
Description: Two numerable sets have the dominance relationship iff their cardinalities have the subset relationship. See also carddom 9232, 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 8656 . 2 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴𝐵))
2 brdom2 7848 . . 3 (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵))
3 cardon 8630 . . . . . . . 8 (card‘𝐴) ∈ On
43onelssi 5739 . . . . . . 7 ((card‘𝐵) ∈ (card‘𝐴) → (card‘𝐵) ⊆ (card‘𝐴))
5 carddomi2 8656 . . . . . . . 8 ((𝐵 ∈ dom card ∧ 𝐴 ∈ dom card) → ((card‘𝐵) ⊆ (card‘𝐴) → 𝐵𝐴))
65ancoms 467 . . . . . . 7 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐵) ⊆ (card‘𝐴) → 𝐵𝐴))
7 domnsym 7948 . . . . . . 7 (𝐵𝐴 → ¬ 𝐴𝐵)
84, 6, 7syl56 35 . . . . . 6 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐵) ∈ (card‘𝐴) → ¬ 𝐴𝐵))
98con2d 127 . . . . 5 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵 → ¬ (card‘𝐵) ∈ (card‘𝐴)))
10 cardon 8630 . . . . . 6 (card‘𝐵) ∈ On
11 ontri1 5660 . . . . . 6 (((card‘𝐴) ∈ On ∧ (card‘𝐵) ∈ On) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ ¬ (card‘𝐵) ∈ (card‘𝐴)))
123, 10, 11mp2an 703 . . . . 5 ((card‘𝐴) ⊆ (card‘𝐵) ↔ ¬ (card‘𝐵) ∈ (card‘𝐴))
139, 12syl6ibr 240 . . . 4 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵 → (card‘𝐴) ⊆ (card‘𝐵)))
14 carden2b 8653 . . . . . 6 (𝐴𝐵 → (card‘𝐴) = (card‘𝐵))
15 eqimss 3619 . . . . . 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 393 . . 3 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((𝐴𝐵𝐴𝐵) → (card‘𝐴) ⊆ (card‘𝐵)))
192, 18syl5bi 230 . 2 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵 → (card‘𝐴) ⊆ (card‘𝐵)))
201, 19impbid 200 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 194  wo 381  wa 382   = wceq 1474  wcel 1976  wss 3539   class class class wbr 4577  dom cdm 5028  Oncon0 5626  cfv 5790  cen 7815  cdom 7816  csdm 7817  cardccrd 8621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-ord 5629  df-on 5630  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-card 8625
This theorem is referenced by:  carduni  8667  carden2  8673  cardsdom2  8674  domtri2  8675  infxpidm2  8700  cardaleph  8772  infenaleph  8774  alephinit  8778  ficardun2  8885  ackbij2  8925  cfflb  8941  fin1a2lem9  9090  carddom  9232  pwfseqlem5  9341  hashdom  12981
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