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Theorem cardne 9605
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 6767 . 2 (𝐴 ∈ (card‘𝐵) → 𝐵 ∈ dom card)
2 cardon 9584 . . . . . . . . . 10 (card‘𝐵) ∈ On
32oneli 6338 . . . . . . . . 9 (𝐴 ∈ (card‘𝐵) → 𝐴 ∈ On)
4 breq1 5070 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
54onintss 6280 . . . . . . . . 9 (𝐴 ∈ On → (𝐴𝐵 {𝑥 ∈ On ∣ 𝑥𝐵} ⊆ 𝐴))
63, 5syl 17 . . . . . . . 8 (𝐴 ∈ (card‘𝐵) → (𝐴𝐵 {𝑥 ∈ On ∣ 𝑥𝐵} ⊆ 𝐴))
76adantl 485 . . . . . . 7 ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → (𝐴𝐵 {𝑥 ∈ On ∣ 𝑥𝐵} ⊆ 𝐴))
8 cardval3 9592 . . . . . . . . 9 (𝐵 ∈ dom card → (card‘𝐵) = {𝑥 ∈ On ∣ 𝑥𝐵})
98sseq1d 3946 . . . . . . . 8 (𝐵 ∈ dom card → ((card‘𝐵) ⊆ 𝐴 {𝑥 ∈ On ∣ 𝑥𝐵} ⊆ 𝐴))
109adantr 484 . . . . . . 7 ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → ((card‘𝐵) ⊆ 𝐴 {𝑥 ∈ On ∣ 𝑥𝐵} ⊆ 𝐴))
117, 10sylibrd 262 . . . . . 6 ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → (𝐴𝐵 → (card‘𝐵) ⊆ 𝐴))
12 ontri1 6264 . . . . . . . 8 (((card‘𝐵) ∈ On ∧ 𝐴 ∈ On) → ((card‘𝐵) ⊆ 𝐴 ↔ ¬ 𝐴 ∈ (card‘𝐵)))
132, 3, 12sylancr 590 . . . . . . 7 (𝐴 ∈ (card‘𝐵) → ((card‘𝐵) ⊆ 𝐴 ↔ ¬ 𝐴 ∈ (card‘𝐵)))
1413adantl 485 . . . . . 6 ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → ((card‘𝐵) ⊆ 𝐴 ↔ ¬ 𝐴 ∈ (card‘𝐵)))
1511, 14sylibd 242 . . . . 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 416 . . 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 209  wa 399  wcel 2111  {crab 3066  wss 3880   cint 4873   class class class wbr 5067  dom cdm 5565  Oncon0 6230  cfv 6397  cen 8643  cardccrd 9575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5206  ax-nul 5213  ax-pr 5336  ax-un 7541
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3422  df-dif 3883  df-un 3885  df-in 3887  df-ss 3897  df-pss 3899  df-nul 4252  df-if 4454  df-pw 4529  df-sn 4556  df-pr 4558  df-tp 4560  df-op 4562  df-uni 4834  df-int 4874  df-br 5068  df-opab 5130  df-mpt 5150  df-tr 5176  df-id 5469  df-eprel 5474  df-po 5482  df-so 5483  df-fr 5523  df-we 5525  df-xp 5571  df-rel 5572  df-cnv 5573  df-co 5574  df-dm 5575  df-rn 5576  df-res 5577  df-ima 5578  df-ord 6233  df-on 6234  df-iota 6355  df-fun 6399  df-fn 6400  df-f 6401  df-fv 6405  df-en 8647  df-card 9579
This theorem is referenced by:  carden2b  9607  cardlim  9612  cardsdomelir  9613
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