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Theorem dif1card 9424
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 8738 . . 3 (𝐴 ∈ Fin → (𝐴 ∖ {𝑋}) ∈ Fin)
2 isfi 8521 . . . 4 ((𝐴 ∖ {𝑋}) ∈ Fin ↔ ∃𝑚 ∈ ω (𝐴 ∖ {𝑋}) ≈ 𝑚)
3 simp3 1130 . . . . . . . . . . 11 ((𝑋𝐴𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → (𝐴 ∖ {𝑋}) ≈ 𝑚)
4 en2sn 8581 . . . . . . . . . . . 12 ((𝑋𝐴𝑚 ∈ ω) → {𝑋} ≈ {𝑚})
543adant3 1124 . . . . . . . . . . 11 ((𝑋𝐴𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → {𝑋} ≈ {𝑚})
6 incom 4175 . . . . . . . . . . . . 13 ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ({𝑋} ∩ (𝐴 ∖ {𝑋}))
7 disjdif 4417 . . . . . . . . . . . . 13 ({𝑋} ∩ (𝐴 ∖ {𝑋})) = ∅
86, 7eqtri 2841 . . . . . . . . . . . 12 ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅
98a1i 11 . . . . . . . . . . 11 ((𝑋𝐴𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅)
10 nnord 7577 . . . . . . . . . . . . . 14 (𝑚 ∈ ω → Ord 𝑚)
11 ordirr 6202 . . . . . . . . . . . . . 14 (Ord 𝑚 → ¬ 𝑚𝑚)
1210, 11syl 17 . . . . . . . . . . . . 13 (𝑚 ∈ ω → ¬ 𝑚𝑚)
13 disjsn 4639 . . . . . . . . . . . . 13 ((𝑚 ∩ {𝑚}) = ∅ ↔ ¬ 𝑚𝑚)
1412, 13sylibr 235 . . . . . . . . . . . 12 (𝑚 ∈ ω → (𝑚 ∩ {𝑚}) = ∅)
15143ad2ant2 1126 . . . . . . . . . . 11 ((𝑋𝐴𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → (𝑚 ∩ {𝑚}) = ∅)
16 unen 8584 . . . . . . . . . . 11 ((((𝐴 ∖ {𝑋}) ≈ 𝑚 ∧ {𝑋} ≈ {𝑚}) ∧ (((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅ ∧ (𝑚 ∩ {𝑚}) = ∅)) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑚 ∪ {𝑚}))
173, 5, 9, 15, 16syl22anc 834 . . . . . . . . . 10 ((𝑋𝐴𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑚 ∪ {𝑚}))
18 difsnid 4735 . . . . . . . . . . . 12 (𝑋𝐴 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴)
19 df-suc 6190 . . . . . . . . . . . . . 14 suc 𝑚 = (𝑚 ∪ {𝑚})
2019eqcomi 2827 . . . . . . . . . . . . 13 (𝑚 ∪ {𝑚}) = suc 𝑚
2120a1i 11 . . . . . . . . . . . 12 (𝑋𝐴 → (𝑚 ∪ {𝑚}) = suc 𝑚)
2218, 21breq12d 5070 . . . . . . . . . . 11 (𝑋𝐴 → (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑚 ∪ {𝑚}) ↔ 𝐴 ≈ suc 𝑚))
23223ad2ant1 1125 . . . . . . . . . 10 ((𝑋𝐴𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑚 ∪ {𝑚}) ↔ 𝐴 ≈ suc 𝑚))
2417, 23mpbid 233 . . . . . . . . 9 ((𝑋𝐴𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → 𝐴 ≈ suc 𝑚)
25 peano2 7591 . . . . . . . . . 10 (𝑚 ∈ ω → suc 𝑚 ∈ ω)
26253ad2ant2 1126 . . . . . . . . 9 ((𝑋𝐴𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → suc 𝑚 ∈ ω)
27 cardennn 9400 . . . . . . . . 9 ((𝐴 ≈ suc 𝑚 ∧ suc 𝑚 ∈ ω) → (card‘𝐴) = suc 𝑚)
2824, 26, 27syl2anc 584 . . . . . . . 8 ((𝑋𝐴𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → (card‘𝐴) = suc 𝑚)
29 cardennn 9400 . . . . . . . . . . 11 (((𝐴 ∖ {𝑋}) ≈ 𝑚𝑚 ∈ ω) → (card‘(𝐴 ∖ {𝑋})) = 𝑚)
3029ancoms 459 . . . . . . . . . 10 ((𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → (card‘(𝐴 ∖ {𝑋})) = 𝑚)
31303adant1 1122 . . . . . . . . 9 ((𝑋𝐴𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → (card‘(𝐴 ∖ {𝑋})) = 𝑚)
32 suceq 6249 . . . . . . . . 9 ((card‘(𝐴 ∖ {𝑋})) = 𝑚 → suc (card‘(𝐴 ∖ {𝑋})) = suc 𝑚)
3331, 32syl 17 . . . . . . . 8 ((𝑋𝐴𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → suc (card‘(𝐴 ∖ {𝑋})) = suc 𝑚)
3428, 33eqtr4d 2856 . . . . . . 7 ((𝑋𝐴𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → (card‘𝐴) = suc (card‘(𝐴 ∖ {𝑋})))
35343expib 1114 . . . . . 6 (𝑋𝐴 → ((𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → (card‘𝐴) = suc (card‘(𝐴 ∖ {𝑋}))))
3635com12 32 . . . . 5 ((𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → (𝑋𝐴 → (card‘𝐴) = suc (card‘(𝐴 ∖ {𝑋}))))
3736rexlimiva 3278 . . . 4 (∃𝑚 ∈ ω (𝐴 ∖ {𝑋}) ≈ 𝑚 → (𝑋𝐴 → (card‘𝐴) = suc (card‘(𝐴 ∖ {𝑋}))))
382, 37sylbi 218 . . 3 ((𝐴 ∖ {𝑋}) ∈ Fin → (𝑋𝐴 → (card‘𝐴) = suc (card‘(𝐴 ∖ {𝑋}))))
391, 38syl 17 . 2 (𝐴 ∈ Fin → (𝑋𝐴 → (card‘𝐴) = suc (card‘(𝐴 ∖ {𝑋}))))
4039imp 407 1 ((𝐴 ∈ Fin ∧ 𝑋𝐴) → (card‘𝐴) = suc (card‘(𝐴 ∖ {𝑋})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wrex 3136  cdif 3930  cun 3931  cin 3932  c0 4288  {csn 4557   class class class wbr 5057  Ord word 6183  suc csuc 6186  cfv 6348  ωcom 7569  cen 8494  Fincfn 8497  cardccrd 9352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-om 7570  df-1o 8091  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-card 9356
This theorem is referenced by:  unidifsnel  30222  unidifsnne  30223
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