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Theorem cardne 9386
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 6698 . 2 (𝐴 ∈ (card‘𝐵) → 𝐵 ∈ dom card)
2 cardon 9365 . . . . . . . . . 10 (card‘𝐵) ∈ On
32oneli 6295 . . . . . . . . 9 (𝐴 ∈ (card‘𝐵) → 𝐴 ∈ On)
4 breq1 5065 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
54onintss 6238 . . . . . . . . 9 (𝐴 ∈ On → (𝐴𝐵 {𝑥 ∈ On ∣ 𝑥𝐵} ⊆ 𝐴))
63, 5syl 17 . . . . . . . 8 (𝐴 ∈ (card‘𝐵) → (𝐴𝐵 {𝑥 ∈ On ∣ 𝑥𝐵} ⊆ 𝐴))
76adantl 482 . . . . . . 7 ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → (𝐴𝐵 {𝑥 ∈ On ∣ 𝑥𝐵} ⊆ 𝐴))
8 cardval3 9373 . . . . . . . . 9 (𝐵 ∈ dom card → (card‘𝐵) = {𝑥 ∈ On ∣ 𝑥𝐵})
98sseq1d 4001 . . . . . . . 8 (𝐵 ∈ dom card → ((card‘𝐵) ⊆ 𝐴 {𝑥 ∈ On ∣ 𝑥𝐵} ⊆ 𝐴))
109adantr 481 . . . . . . 7 ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → ((card‘𝐵) ⊆ 𝐴 {𝑥 ∈ On ∣ 𝑥𝐵} ⊆ 𝐴))
117, 10sylibrd 260 . . . . . 6 ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → (𝐴𝐵 → (card‘𝐵) ⊆ 𝐴))
12 ontri1 6222 . . . . . . . 8 (((card‘𝐵) ∈ On ∧ 𝐴 ∈ On) → ((card‘𝐵) ⊆ 𝐴 ↔ ¬ 𝐴 ∈ (card‘𝐵)))
132, 3, 12sylancr 587 . . . . . . 7 (𝐴 ∈ (card‘𝐵) → ((card‘𝐵) ⊆ 𝐴 ↔ ¬ 𝐴 ∈ (card‘𝐵)))
1413adantl 482 . . . . . 6 ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → ((card‘𝐵) ⊆ 𝐴 ↔ ¬ 𝐴 ∈ (card‘𝐵)))
1511, 14sylibd 240 . . . . 5 ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → (𝐴𝐵 → ¬ 𝐴 ∈ (card‘𝐵)))
1615con2d 136 . . . 4 ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → (𝐴 ∈ (card‘𝐵) → ¬ 𝐴𝐵))
1716ex 413 . . 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 207  wa 396  wcel 2107  {crab 3146  wss 3939   cint 4873   class class class wbr 5062  dom cdm 5553  Oncon0 6188  cfv 6351  cen 8498  cardccrd 9356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-ord 6191  df-on 6192  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-fv 6359  df-en 8502  df-card 9360
This theorem is referenced by:  carden2b  9388  cardlim  9393  cardsdomelir  9394
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