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Theorem cardne 8674
 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 6130 . 2 (𝐴 ∈ (card‘𝐵) → 𝐵 ∈ dom card)
2 cardon 8653 . . . . . . . . . 10 (card‘𝐵) ∈ On
32oneli 5752 . . . . . . . . 9 (𝐴 ∈ (card‘𝐵) → 𝐴 ∈ On)
4 breq1 4586 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
54onintss 5692 . . . . . . . . 9 (𝐴 ∈ On → (𝐴𝐵 {𝑥 ∈ On ∣ 𝑥𝐵} ⊆ 𝐴))
63, 5syl 17 . . . . . . . 8 (𝐴 ∈ (card‘𝐵) → (𝐴𝐵 {𝑥 ∈ On ∣ 𝑥𝐵} ⊆ 𝐴))
76adantl 481 . . . . . . 7 ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → (𝐴𝐵 {𝑥 ∈ On ∣ 𝑥𝐵} ⊆ 𝐴))
8 cardval3 8661 . . . . . . . . 9 (𝐵 ∈ dom card → (card‘𝐵) = {𝑥 ∈ On ∣ 𝑥𝐵})
98sseq1d 3595 . . . . . . . 8 (𝐵 ∈ dom card → ((card‘𝐵) ⊆ 𝐴 {𝑥 ∈ On ∣ 𝑥𝐵} ⊆ 𝐴))
109adantr 480 . . . . . . 7 ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → ((card‘𝐵) ⊆ 𝐴 {𝑥 ∈ On ∣ 𝑥𝐵} ⊆ 𝐴))
117, 10sylibrd 248 . . . . . 6 ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → (𝐴𝐵 → (card‘𝐵) ⊆ 𝐴))
12 ontri1 5674 . . . . . . . 8 (((card‘𝐵) ∈ On ∧ 𝐴 ∈ On) → ((card‘𝐵) ⊆ 𝐴 ↔ ¬ 𝐴 ∈ (card‘𝐵)))
132, 3, 12sylancr 694 . . . . . . 7 (𝐴 ∈ (card‘𝐵) → ((card‘𝐵) ⊆ 𝐴 ↔ ¬ 𝐴 ∈ (card‘𝐵)))
1413adantl 481 . . . . . 6 ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → ((card‘𝐵) ⊆ 𝐴 ↔ ¬ 𝐴 ∈ (card‘𝐵)))
1511, 14sylibd 228 . . . . 5 ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → (𝐴𝐵 → ¬ 𝐴 ∈ (card‘𝐵)))
1615con2d 128 . . . 4 ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → (𝐴 ∈ (card‘𝐵) → ¬ 𝐴𝐵))
1716ex 449 . . 3 (𝐵 ∈ dom card → (𝐴 ∈ (card‘𝐵) → (𝐴 ∈ (card‘𝐵) → ¬ 𝐴𝐵)))
1817pm2.43d 51 . 2 (𝐵 ∈ dom card → (𝐴 ∈ (card‘𝐵) → ¬ 𝐴𝐵))
191, 18mpcom 37 1 (𝐴 ∈ (card‘𝐵) → ¬ 𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∈ wcel 1977  {crab 2900   ⊆ wss 3540  ∩ cint 4410   class class class wbr 4583  dom cdm 5038  Oncon0 5640  ‘cfv 5804   ≈ cen 7838  cardccrd 8644 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-en 7842  df-card 8648 This theorem is referenced by:  carden2b  8676  cardlim  8681  cardsdomelir  8682
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