Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  domtri2 Structured version   Visualization version   GIF version

Theorem domtri2 9127
 Description: Trichotomy of dominance for numerable sets (does not use AC). (Contributed by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
domtri2 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))

Proof of Theorem domtri2
StepHypRef Expression
1 carddom2 9115 . 2 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ 𝐴𝐵))
2 cardon 9082 . . . 4 (card‘𝐴) ∈ On
3 cardon 9082 . . . 4 (card‘𝐵) ∈ On
4 ontri1 5996 . . . 4 (((card‘𝐴) ∈ On ∧ (card‘𝐵) ∈ On) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ ¬ (card‘𝐵) ∈ (card‘𝐴)))
52, 3, 4mp2an 685 . . 3 ((card‘𝐴) ⊆ (card‘𝐵) ↔ ¬ (card‘𝐵) ∈ (card‘𝐴))
6 cardsdom2 9126 . . . . 5 ((𝐵 ∈ dom card ∧ 𝐴 ∈ dom card) → ((card‘𝐵) ∈ (card‘𝐴) ↔ 𝐵𝐴))
76ancoms 452 . . . 4 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐵) ∈ (card‘𝐴) ↔ 𝐵𝐴))
87notbid 310 . . 3 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (¬ (card‘𝐵) ∈ (card‘𝐴) ↔ ¬ 𝐵𝐴))
95, 8syl5bb 275 . 2 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ ¬ 𝐵𝐴))
101, 9bitr3d 273 1 ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   ∈ wcel 2166   ⊆ wss 3797   class class class wbr 4872  dom cdm 5341  Oncon0 5962  ‘cfv 6122   ≼ cdom 8219   ≺ csdm 8220  cardccrd 9073 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 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pow 5064  ax-pr 5126  ax-un 7208 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 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ne 2999  df-ral 3121  df-rex 3122  df-rab 3125  df-v 3415  df-sbc 3662  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-pss 3813  df-nul 4144  df-if 4306  df-pw 4379  df-sn 4397  df-pr 4399  df-tp 4401  df-op 4403  df-uni 4658  df-int 4697  df-br 4873  df-opab 4935  df-mpt 4952  df-tr 4975  df-id 5249  df-eprel 5254  df-po 5262  df-so 5263  df-fr 5300  df-we 5302  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-ima 5354  df-ord 5965  df-on 5966  df-iota 6085  df-fun 6124  df-fn 6125  df-f 6126  df-f1 6127  df-fo 6128  df-f1o 6129  df-fv 6130  df-er 8008  df-en 8222  df-dom 8223  df-sdom 8224  df-card 9077 This theorem is referenced by:  fidomtri  9131  harsdom  9133  infdif  9345  infdif2  9346  infunsdom1  9349  infunsdom  9350  infxp  9351  domtri  9692  canthp1lem2  9789  pwfseqlem4a  9797  pwfseqlem4  9798  gchaleph  9807  numinfctb  38515
 Copyright terms: Public domain W3C validator