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

Theorem cardne 10008
Description: No member of a cardinal number of a set is equinumerous to the set. Proposition 10.6(2) of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 9-Jan-2013.)
Assertion
Ref Expression
cardne (𝐴 ∈ (card‘𝐵) → ¬ 𝐴𝐵)

Proof of Theorem cardne
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elfvdm 6938 . 2 (𝐴 ∈ (card‘𝐵) → 𝐵 ∈ dom card)
2 cardon 9987 . . . . . . . . . 10 (card‘𝐵) ∈ On
32oneli 6490 . . . . . . . . 9 (𝐴 ∈ (card‘𝐵) → 𝐴 ∈ On)
4 breq1 5156 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
54onintss 6427 . . . . . . . . 9 (𝐴 ∈ On → (𝐴𝐵 {𝑥 ∈ On ∣ 𝑥𝐵} ⊆ 𝐴))
63, 5syl 17 . . . . . . . 8 (𝐴 ∈ (card‘𝐵) → (𝐴𝐵 {𝑥 ∈ On ∣ 𝑥𝐵} ⊆ 𝐴))
76adantl 480 . . . . . . 7 ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → (𝐴𝐵 {𝑥 ∈ On ∣ 𝑥𝐵} ⊆ 𝐴))
8 cardval3 9995 . . . . . . . . 9 (𝐵 ∈ dom card → (card‘𝐵) = {𝑥 ∈ On ∣ 𝑥𝐵})
98sseq1d 4011 . . . . . . . 8 (𝐵 ∈ dom card → ((card‘𝐵) ⊆ 𝐴 {𝑥 ∈ On ∣ 𝑥𝐵} ⊆ 𝐴))
109adantr 479 . . . . . . 7 ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → ((card‘𝐵) ⊆ 𝐴 {𝑥 ∈ On ∣ 𝑥𝐵} ⊆ 𝐴))
117, 10sylibrd 258 . . . . . 6 ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → (𝐴𝐵 → (card‘𝐵) ⊆ 𝐴))
12 ontri1 6410 . . . . . . . 8 (((card‘𝐵) ∈ On ∧ 𝐴 ∈ On) → ((card‘𝐵) ⊆ 𝐴 ↔ ¬ 𝐴 ∈ (card‘𝐵)))
132, 3, 12sylancr 585 . . . . . . 7 (𝐴 ∈ (card‘𝐵) → ((card‘𝐵) ⊆ 𝐴 ↔ ¬ 𝐴 ∈ (card‘𝐵)))
1413adantl 480 . . . . . 6 ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → ((card‘𝐵) ⊆ 𝐴 ↔ ¬ 𝐴 ∈ (card‘𝐵)))
1511, 14sylibd 238 . . . . 5 ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → (𝐴𝐵 → ¬ 𝐴 ∈ (card‘𝐵)))
1615con2d 134 . . . 4 ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → (𝐴 ∈ (card‘𝐵) → ¬ 𝐴𝐵))
1716ex 411 . . 3 (𝐵 ∈ dom card → (𝐴 ∈ (card‘𝐵) → (𝐴 ∈ (card‘𝐵) → ¬ 𝐴𝐵)))
1817pm2.43d 53 . 2 (𝐵 ∈ dom card → (𝐴 ∈ (card‘𝐵) → ¬ 𝐴𝐵))
191, 18mpcom 38 1 (𝐴 ∈ (card‘𝐵) → ¬ 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wcel 2099  {crab 3419  wss 3947   cint 4954   class class class wbr 5153  dom cdm 5682  Oncon0 6376  cfv 6554  cen 8971  cardccrd 9978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6379  df-on 6380  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-fv 6562  df-en 8975  df-card 9982
This theorem is referenced by:  carden2b  10010  cardlim  10015  cardsdomelir  10016
  Copyright terms: Public domain W3C validator