Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cardsdom2 Structured version   Visualization version   GIF version

Theorem cardsdom2 8759
 Description: A numerable set is strictly dominated by another iff their cardinalities are strictly ordered. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
cardsdom2 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ∈ (card‘𝐵) ↔ 𝐴𝐵))

Proof of Theorem cardsdom2
StepHypRef Expression
1 carddom2 8748 . . 3 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ 𝐴𝐵))
2 carden2 8758 . . . 4 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) = (card‘𝐵) ↔ 𝐴𝐵))
32necon3abid 2832 . . 3 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ≠ (card‘𝐵) ↔ ¬ 𝐴𝐵))
41, 3anbi12d 746 . 2 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (((card‘𝐴) ⊆ (card‘𝐵) ∧ (card‘𝐴) ≠ (card‘𝐵)) ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐵)))
5 cardon 8715 . . 3 (card‘𝐴) ∈ On
6 cardon 8715 . . 3 (card‘𝐵) ∈ On
7 onelpss 5726 . . 3 (((card‘𝐴) ∈ On ∧ (card‘𝐵) ∈ On) → ((card‘𝐴) ∈ (card‘𝐵) ↔ ((card‘𝐴) ⊆ (card‘𝐵) ∧ (card‘𝐴) ≠ (card‘𝐵))))
85, 6, 7mp2an 707 . 2 ((card‘𝐴) ∈ (card‘𝐵) ↔ ((card‘𝐴) ⊆ (card‘𝐵) ∧ (card‘𝐴) ≠ (card‘𝐵)))
9 brsdom 7923 . 2 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐵))
104, 8, 93bitr4g 303 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   ∧ wa 384   ∈ wcel 1992   ≠ wne 2796   ⊆ wss 3560   class class class wbr 4618  dom cdm 5079  Oncon0 5685  ‘cfv 5850   ≈ cen 7897   ≼ cdom 7898   ≺ csdm 7899  cardccrd 8706 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-ord 5688  df-on 5689  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-card 8710 This theorem is referenced by:  domtri2  8760  nnsdomel  8761  indcardi  8809  sdom2en01  9069  cardsdom  9322  smobeth  9353  hargch  9440
 Copyright terms: Public domain W3C validator