Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  carddom2 Structured version   Visualization version   GIF version

Theorem carddom2 8686
 Description: Two numerable sets have the dominance relationship iff their cardinalities have the subset relationship. See also carddom 9255, 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 8679 . 2 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝐵) → 𝐴𝐵))
2 brdom2 7871 . . 3 (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵))
3 cardon 8653 . . . . . . . 8 (card‘𝐴) ∈ On
43onelssi 5753 . . . . . . 7 ((card‘𝐵) ∈ (card‘𝐴) → (card‘𝐵) ⊆ (card‘𝐴))
5 carddomi2 8679 . . . . . . . 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 7971 . . . . . . 7 (𝐵𝐴 → ¬ 𝐴𝐵)
84, 6, 7syl56 35 . . . . . 6 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐵) ∈ (card‘𝐴) → ¬ 𝐴𝐵))
98con2d 128 . . . . 5 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵 → ¬ (card‘𝐵) ∈ (card‘𝐴)))
10 cardon 8653 . . . . . 6 (card‘𝐵) ∈ On
11 ontri1 5674 . . . . . 6 (((card‘𝐴) ∈ On ∧ (card‘𝐵) ∈ On) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ ¬ (card‘𝐵) ∈ (card‘𝐴)))
123, 10, 11mp2an 704 . . . . 5 ((card‘𝐴) ⊆ (card‘𝐵) ↔ ¬ (card‘𝐵) ∈ (card‘𝐴))
139, 12syl6ibr 241 . . . 4 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵 → (card‘𝐴) ⊆ (card‘𝐵)))
14 carden2b 8676 . . . . . 6 (𝐴𝐵 → (card‘𝐴) = (card‘𝐵))
15 eqimss 3620 . . . . . 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 231 . 2 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵 → (card‘𝐴) ⊆ (card‘𝐵)))
201, 19impbid 201 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 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540   class class class wbr 4583  dom cdm 5038  Oncon0 5640  ‘cfv 5804   ≈ cen 7838   ≼ cdom 7839   ≺ csdm 7840  cardccrd 8644 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-card 8648 This theorem is referenced by:  carduni  8690  carden2  8696  cardsdom2  8697  domtri2  8698  infxpidm2  8723  cardaleph  8795  infenaleph  8797  alephinit  8801  ficardun2  8908  ackbij2  8948  cfflb  8964  fin1a2lem9  9113  carddom  9255  pwfseqlem5  9364  hashdom  13029
 Copyright terms: Public domain W3C validator