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Theorem domtri2 9984
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 ∧ 𝐡 ∈ dom card) β†’ (𝐴 β‰Ό 𝐡 ↔ Β¬ 𝐡 β‰Ί 𝐴))

Proof of Theorem domtri2
StepHypRef Expression
1 carddom2 9972 . 2 ((𝐴 ∈ dom card ∧ 𝐡 ∈ dom card) β†’ ((cardβ€˜π΄) βŠ† (cardβ€˜π΅) ↔ 𝐴 β‰Ό 𝐡))
2 cardon 9939 . . . 4 (cardβ€˜π΄) ∈ On
3 cardon 9939 . . . 4 (cardβ€˜π΅) ∈ On
4 ontri1 6399 . . . 4 (((cardβ€˜π΄) ∈ On ∧ (cardβ€˜π΅) ∈ On) β†’ ((cardβ€˜π΄) βŠ† (cardβ€˜π΅) ↔ Β¬ (cardβ€˜π΅) ∈ (cardβ€˜π΄)))
52, 3, 4mp2an 691 . . 3 ((cardβ€˜π΄) βŠ† (cardβ€˜π΅) ↔ Β¬ (cardβ€˜π΅) ∈ (cardβ€˜π΄))
6 cardsdom2 9983 . . . . 5 ((𝐡 ∈ dom card ∧ 𝐴 ∈ dom card) β†’ ((cardβ€˜π΅) ∈ (cardβ€˜π΄) ↔ 𝐡 β‰Ί 𝐴))
76ancoms 460 . . . 4 ((𝐴 ∈ dom card ∧ 𝐡 ∈ dom card) β†’ ((cardβ€˜π΅) ∈ (cardβ€˜π΄) ↔ 𝐡 β‰Ί 𝐴))
87notbid 318 . . 3 ((𝐴 ∈ dom card ∧ 𝐡 ∈ dom card) β†’ (Β¬ (cardβ€˜π΅) ∈ (cardβ€˜π΄) ↔ Β¬ 𝐡 β‰Ί 𝐴))
95, 8bitrid 283 . 2 ((𝐴 ∈ dom card ∧ 𝐡 ∈ dom card) β†’ ((cardβ€˜π΄) βŠ† (cardβ€˜π΅) ↔ Β¬ 𝐡 β‰Ί 𝐴))
101, 9bitr3d 281 1 ((𝐴 ∈ dom card ∧ 𝐡 ∈ dom card) β†’ (𝐴 β‰Ό 𝐡 ↔ Β¬ 𝐡 β‰Ί 𝐴))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∈ wcel 2107   βŠ† wss 3949   class class class wbr 5149  dom cdm 5677  Oncon0 6365  β€˜cfv 6544   β‰Ό cdom 8937   β‰Ί csdm 8938  cardccrd 9930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-card 9934
This theorem is referenced by:  fidomtri  9988  harsdom  9990  infdif  10204  infdif2  10205  infunsdom1  10208  infunsdom  10209  infxp  10210  domtri  10551  canthp1lem2  10648  pwfseqlem4a  10656  pwfseqlem4  10657  gchaleph  10666  numinfctb  41845
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