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Theorem carddom2 8841
Description: Two numerable sets have the dominance relationship iff their cardinalities have the subset relationship. See also carddom 9414, 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 8834 . 2 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴𝐵))
2 brdom2 8027 . . 3 (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵))
3 cardon 8808 . . . . . . . 8 (card‘𝐴) ∈ On
43onelssi 5874 . . . . . . 7 ((card‘𝐵) ∈ (card‘𝐴) → (card‘𝐵) ⊆ (card‘𝐴))
5 carddomi2 8834 . . . . . . . 8 ((𝐵 ∈ dom card ∧ 𝐴 ∈ dom card) → ((card‘𝐵) ⊆ (card‘𝐴) → 𝐵𝐴))
65ancoms 468 . . . . . . 7 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐵) ⊆ (card‘𝐴) → 𝐵𝐴))
7 domnsym 8127 . . . . . . 7 (𝐵𝐴 → ¬ 𝐴𝐵)
84, 6, 7syl56 36 . . . . . 6 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐵) ∈ (card‘𝐴) → ¬ 𝐴𝐵))
98con2d 129 . . . . 5 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵 → ¬ (card‘𝐵) ∈ (card‘𝐴)))
10 cardon 8808 . . . . . 6 (card‘𝐵) ∈ On
11 ontri1 5795 . . . . . 6 (((card‘𝐴) ∈ On ∧ (card‘𝐵) ∈ On) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ ¬ (card‘𝐵) ∈ (card‘𝐴)))
123, 10, 11mp2an 708 . . . . 5 ((card‘𝐴) ⊆ (card‘𝐵) ↔ ¬ (card‘𝐵) ∈ (card‘𝐴))
139, 12syl6ibr 242 . . . 4 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵 → (card‘𝐴) ⊆ (card‘𝐵)))
14 carden2b 8831 . . . . . 6 (𝐴𝐵 → (card‘𝐴) = (card‘𝐵))
15 eqimss 3690 . . . . . 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 394 . . 3 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((𝐴𝐵𝐴𝐵) → (card‘𝐴) ⊆ (card‘𝐵)))
192, 18syl5bi 232 . 2 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵 → (card‘𝐴) ⊆ (card‘𝐵)))
201, 19impbid 202 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 196  wo 382  wa 383   = wceq 1523  wcel 2030  wss 3607   class class class wbr 4685  dom cdm 5143  Oncon0 5761  cfv 5926  cen 7994  cdom 7995  csdm 7996  cardccrd 8799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-card 8803
This theorem is referenced by:  carduni  8845  carden2  8851  cardsdom2  8852  domtri2  8853  infxpidm2  8878  cardaleph  8950  infenaleph  8952  alephinit  8956  ficardun2  9063  ackbij2  9103  cfflb  9119  fin1a2lem9  9268  carddom  9414  pwfseqlem5  9523  hashdom  13206
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