MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dif1card Structured version   Visualization version   GIF version

Theorem dif1card 10050
Description: The cardinality of a nonempty finite set is one greater than the cardinality of the set with one element removed. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 2-Feb-2013.)
Assertion
Ref Expression
dif1card ((𝐴 ∈ Fin ∧ 𝑋𝐴) → (card‘𝐴) = suc (card‘(𝐴 ∖ {𝑋})))

Proof of Theorem dif1card
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 diffi 9215 . . 3 (𝐴 ∈ Fin → (𝐴 ∖ {𝑋}) ∈ Fin)
2 isfi 9016 . . . 4 ((𝐴 ∖ {𝑋}) ∈ Fin ↔ ∃𝑚 ∈ ω (𝐴 ∖ {𝑋}) ≈ 𝑚)
3 simp3 1139 . . . . . . . . . . 11 ((𝑋𝐴𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → (𝐴 ∖ {𝑋}) ≈ 𝑚)
4 en2sn 9081 . . . . . . . . . . . 12 ((𝑋𝐴𝑚 ∈ ω) → {𝑋} ≈ {𝑚})
543adant3 1133 . . . . . . . . . . 11 ((𝑋𝐴𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → {𝑋} ≈ {𝑚})
6 disjdifr 4473 . . . . . . . . . . . 12 ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅
76a1i 11 . . . . . . . . . . 11 ((𝑋𝐴𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅)
8 nnord 7895 . . . . . . . . . . . . . 14 (𝑚 ∈ ω → Ord 𝑚)
9 ordirr 6402 . . . . . . . . . . . . . 14 (Ord 𝑚 → ¬ 𝑚𝑚)
108, 9syl 17 . . . . . . . . . . . . 13 (𝑚 ∈ ω → ¬ 𝑚𝑚)
11 disjsn 4711 . . . . . . . . . . . . 13 ((𝑚 ∩ {𝑚}) = ∅ ↔ ¬ 𝑚𝑚)
1210, 11sylibr 234 . . . . . . . . . . . 12 (𝑚 ∈ ω → (𝑚 ∩ {𝑚}) = ∅)
13123ad2ant2 1135 . . . . . . . . . . 11 ((𝑋𝐴𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → (𝑚 ∩ {𝑚}) = ∅)
14 unen 9086 . . . . . . . . . . 11 ((((𝐴 ∖ {𝑋}) ≈ 𝑚 ∧ {𝑋} ≈ {𝑚}) ∧ (((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅ ∧ (𝑚 ∩ {𝑚}) = ∅)) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑚 ∪ {𝑚}))
153, 5, 7, 13, 14syl22anc 839 . . . . . . . . . 10 ((𝑋𝐴𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑚 ∪ {𝑚}))
16 difsnid 4810 . . . . . . . . . . . 12 (𝑋𝐴 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴)
17 df-suc 6390 . . . . . . . . . . . . . 14 suc 𝑚 = (𝑚 ∪ {𝑚})
1817eqcomi 2746 . . . . . . . . . . . . 13 (𝑚 ∪ {𝑚}) = suc 𝑚
1918a1i 11 . . . . . . . . . . . 12 (𝑋𝐴 → (𝑚 ∪ {𝑚}) = suc 𝑚)
2016, 19breq12d 5156 . . . . . . . . . . 11 (𝑋𝐴 → (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑚 ∪ {𝑚}) ↔ 𝐴 ≈ suc 𝑚))
21203ad2ant1 1134 . . . . . . . . . 10 ((𝑋𝐴𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑚 ∪ {𝑚}) ↔ 𝐴 ≈ suc 𝑚))
2215, 21mpbid 232 . . . . . . . . 9 ((𝑋𝐴𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → 𝐴 ≈ suc 𝑚)
23 peano2 7912 . . . . . . . . . 10 (𝑚 ∈ ω → suc 𝑚 ∈ ω)
24233ad2ant2 1135 . . . . . . . . 9 ((𝑋𝐴𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → suc 𝑚 ∈ ω)
25 cardennn 10023 . . . . . . . . 9 ((𝐴 ≈ suc 𝑚 ∧ suc 𝑚 ∈ ω) → (card‘𝐴) = suc 𝑚)
2622, 24, 25syl2anc 584 . . . . . . . 8 ((𝑋𝐴𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → (card‘𝐴) = suc 𝑚)
27 cardennn 10023 . . . . . . . . . . 11 (((𝐴 ∖ {𝑋}) ≈ 𝑚𝑚 ∈ ω) → (card‘(𝐴 ∖ {𝑋})) = 𝑚)
2827ancoms 458 . . . . . . . . . 10 ((𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → (card‘(𝐴 ∖ {𝑋})) = 𝑚)
29283adant1 1131 . . . . . . . . 9 ((𝑋𝐴𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → (card‘(𝐴 ∖ {𝑋})) = 𝑚)
30 suceq 6450 . . . . . . . . 9 ((card‘(𝐴 ∖ {𝑋})) = 𝑚 → suc (card‘(𝐴 ∖ {𝑋})) = suc 𝑚)
3129, 30syl 17 . . . . . . . 8 ((𝑋𝐴𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → suc (card‘(𝐴 ∖ {𝑋})) = suc 𝑚)
3226, 31eqtr4d 2780 . . . . . . 7 ((𝑋𝐴𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → (card‘𝐴) = suc (card‘(𝐴 ∖ {𝑋})))
33323expib 1123 . . . . . 6 (𝑋𝐴 → ((𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → (card‘𝐴) = suc (card‘(𝐴 ∖ {𝑋}))))
3433com12 32 . . . . 5 ((𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → (𝑋𝐴 → (card‘𝐴) = suc (card‘(𝐴 ∖ {𝑋}))))
3534rexlimiva 3147 . . . 4 (∃𝑚 ∈ ω (𝐴 ∖ {𝑋}) ≈ 𝑚 → (𝑋𝐴 → (card‘𝐴) = suc (card‘(𝐴 ∖ {𝑋}))))
362, 35sylbi 217 . . 3 ((𝐴 ∖ {𝑋}) ∈ Fin → (𝑋𝐴 → (card‘𝐴) = suc (card‘(𝐴 ∖ {𝑋}))))
371, 36syl 17 . 2 (𝐴 ∈ Fin → (𝑋𝐴 → (card‘𝐴) = suc (card‘(𝐴 ∖ {𝑋}))))
3837imp 406 1 ((𝐴 ∈ Fin ∧ 𝑋𝐴) → (card‘𝐴) = suc (card‘(𝐴 ∖ {𝑋})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wrex 3070  cdif 3948  cun 3949  cin 3950  c0 4333  {csn 4626   class class class wbr 5143  Ord word 6383  suc csuc 6386  cfv 6561  ωcom 7887  cen 8982  Fincfn 8985  cardccrd 9975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979
This theorem is referenced by:  unidifsnel  32553  unidifsnne  32554
  Copyright terms: Public domain W3C validator