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Theorem carddom2 9116
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.)
Assertion
Ref Expression
carddom2 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ 𝐴𝐵))

Proof of Theorem carddom2
StepHypRef Expression
1 carddomi2 9109 . 2 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴𝐵))
2 brdom2 8252 . . 3 (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵))
3 cardon 9083 . . . . . . . 8 (card‘𝐴) ∈ On
43onelssi 6071 . . . . . . 7 ((card‘𝐵) ∈ (card‘𝐴) → (card‘𝐵) ⊆ (card‘𝐴))
5 carddomi2 9109 . . . . . . . 8 ((𝐵 ∈ dom card ∧ 𝐴 ∈ dom card) → ((card‘𝐵) ⊆ (card‘𝐴) → 𝐵𝐴))
65ancoms 452 . . . . . . 7 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐵) ⊆ (card‘𝐴) → 𝐵𝐴))
7 domnsym 8355 . . . . . . 7 (𝐵𝐴 → ¬ 𝐴𝐵)
84, 6, 7syl56 36 . . . . . 6 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐵) ∈ (card‘𝐴) → ¬ 𝐴𝐵))
98con2d 132 . . . . 5 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵 → ¬ (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)))
123, 10, 11mp2an 685 . . . . 5 ((card‘𝐴) ⊆ (card‘𝐵) ↔ ¬ (card‘𝐵) ∈ (card‘𝐴))
139, 12syl6ibr 244 . . . 4 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵 → (card‘𝐴) ⊆ (card‘𝐵)))
14 carden2b 9106 . . . . . 6 (𝐴𝐵 → (card‘𝐴) = (card‘𝐵))
15 eqimss 3882 . . . . . 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 892 . . 3 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((𝐴𝐵𝐴𝐵) → (card‘𝐴) ⊆ (card‘𝐵)))
192, 18syl5bi 234 . 2 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵 → (card‘𝐴) ⊆ (card‘𝐵)))
201, 19impbid 204 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 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
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