Theorem cardsucnn 9743

Description: The cardinality of the successor of a finite ordinal (natural number). This theorem does not hold for infinite ordinals; see cardsucinf 9742. (Contributed by NM, 7-Nov-2008.)

Assertion

Ref	Expression
cardsucnn	⊢ (𝐴 ∈ ω → (card‘suc 𝐴) = suc (card‘𝐴))

Proof of Theorem cardsucnn

Step	Hyp	Ref	Expression
1		peano2 7737	. . 3 ⊢ (𝐴 ∈ ω → suc 𝐴 ∈ ω)
2		cardnn 9721	. . 3 ⊢ (suc 𝐴 ∈ ω → (card‘suc 𝐴) = suc 𝐴)
3	1, 2	syl 17	. 2 ⊢ (𝐴 ∈ ω → (card‘suc 𝐴) = suc 𝐴)
4		cardnn 9721	. . 3 ⊢ (𝐴 ∈ ω → (card‘𝐴) = 𝐴)
5		suceq 6331	. . 3 ⊢ ((card‘𝐴) = 𝐴 → suc (card‘𝐴) = suc 𝐴)
6	4, 5	syl 17	. 2 ⊢ (𝐴 ∈ ω → suc (card‘𝐴) = suc 𝐴)
7	3, 6	eqtr4d 2781	1 ⊢ (𝐴 ∈ ω → (card‘suc 𝐴) = suc (card‘𝐴))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  suc csuc 6268  ‘cfv 6433  ωcom 7712  cardccrd 9693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-om 7713  df-1o 8297  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-card 9697

This theorem is referenced by: (None)