Theorem cardne 3637

Description: No member of a cardinal number of a set is equinumerous to the set. Proposition 10.6(2) of [TakeutiZaring] p. 85.

Assertion

Ref	Expression
cardne	⊢ (A ∈ (card ‘B) → ¬ A ≈ B)

Proof of Theorem cardne

Step	Hyp	Ref	Expression
1		cardon 3634	. . 3 ⊢ (card ‘B) ∈ On
2	1	onel 2346	. 2 ⊢ (A ∈ (card ‘B) → A ∈ On)
3		breq1 2065	. . . . . 6 ⊢ (x = A → (x ≈ B ↔ A ≈ B))
4	3	onintss 2266	. . . . 5 ⊢ (A ∈ On → (A ≈ B → ∩{x ∈ On∣x ≈ B} ⊆ A))
5		cardval 3633	. . . . . 6 ⊢ (card ‘B) = ∩{x ∈ On∣x ≈ B}
6	5	sseq1i 1524	. . . . 5 ⊢ ((card ‘B) ⊆ A ↔ ∩{x ∈ On∣x ≈ B} ⊆ A)
7	4, 6	syl6ibr 186	. . . 4 ⊢ (A ∈ On → (A ≈ B → (card ‘B) ⊆ A))
8		ontri1 2232	. . . . 5 ⊢ (((card ‘B) ∈ On ∧ A ∈ On) → ((card ‘B) ⊆ A ↔ ¬ A ∈ (card ‘B)))
9	1, 8	mpan 518	. . . 4 ⊢ (A ∈ On → ((card ‘B) ⊆ A ↔ ¬ A ∈ (card ‘B)))
10	7, 9	sylibd 177	. . 3 ⊢ (A ∈ On → (A ≈ B → ¬ A ∈ (card ‘B)))
11	10	con2d 83	. 2 ⊢ (A ∈ On → (A ∈ (card ‘B) → ¬ A ≈ B))
12	2, 11	mpcom 49	1 ⊢ (A ∈ (card ‘B) → ¬ A ≈ B)

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∈ wcel 1092  {crab 1204   ⊆ wss 1487  ∩cint 1965   class class class wbr 2054  Oncon0 2199   ‘cfv 2422   ≈ cen 3271  cardccrd 3620

This theorem is referenced by:  carden 3638  cardlim 3657  cardsdomel 3658

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-ac 1080

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-suc 2205  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-en 3274  df-card 3623

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