Theorem dif1card 9922

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 ∧ 𝑋 ∈ 𝐴) → (card‘𝐴) = suc (card‘(𝐴 ∖ {𝑋})))

Proof of Theorem dif1card

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		diffi 9101	. . 3 ⊢ (𝐴 ∈ Fin → (𝐴 ∖ {𝑋}) ∈ Fin)
2		isfi 8914	. . . 4 ⊢ ((𝐴 ∖ {𝑋}) ∈ Fin ↔ ∃𝑚 ∈ ω (𝐴 ∖ {𝑋}) ≈ 𝑚)
3		simp3 1139	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → (𝐴 ∖ {𝑋}) ≈ 𝑚)
4		en2sn 8980	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω) → {𝑋} ≈ {𝑚})
5	4	3adant3 1133	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → {𝑋} ≈ {𝑚})
6		disjdifr 4424	. . . . . . . . . . . 12 ⊢ ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅
7	6	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅)
8		nnord 7816	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ω → Ord 𝑚)
9		ordirr 6334	. . . . . . . . . . . . . 14 ⊢ (Ord 𝑚 → ¬ 𝑚 ∈ 𝑚)
10	8, 9	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ω → ¬ 𝑚 ∈ 𝑚)
11		disjsn 4667	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∩ {𝑚}) = ∅ ↔ ¬ 𝑚 ∈ 𝑚)
12	10, 11	sylibr 234	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ω → (𝑚 ∩ {𝑚}) = ∅)
13	12	3ad2ant2 1135	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → (𝑚 ∩ {𝑚}) = ∅)
14		unen 8984	. . . . . . . . . . 11 ⊢ ((((𝐴 ∖ {𝑋}) ≈ 𝑚 ∧ {𝑋} ≈ {𝑚}) ∧ (((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅ ∧ (𝑚 ∩ {𝑚}) = ∅)) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑚 ∪ {𝑚}))
15	3, 5, 7, 13, 14	syl22anc 839	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑚 ∪ {𝑚}))
16		difsnid 4765	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝐴 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴)
17		df-suc 6322	. . . . . . . . . . . . . 14 ⊢ suc 𝑚 = (𝑚 ∪ {𝑚})
18	17	eqcomi 2744	. . . . . . . . . . . . 13 ⊢ (𝑚 ∪ {𝑚}) = suc 𝑚
19	18	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝐴 → (𝑚 ∪ {𝑚}) = suc 𝑚)
20	16, 19	breq12d 5110	. . . . . . . . . . 11 ⊢ (𝑋 ∈ 𝐴 → (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑚 ∪ {𝑚}) ↔ 𝐴 ≈ suc 𝑚))
21	20	3ad2ant1 1134	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑚 ∪ {𝑚}) ↔ 𝐴 ≈ suc 𝑚))
22	15, 21	mpbid 232	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → 𝐴 ≈ suc 𝑚)
23		peano2 7832	. . . . . . . . . 10 ⊢ (𝑚 ∈ ω → suc 𝑚 ∈ ω)
24	23	3ad2ant2 1135	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → suc 𝑚 ∈ ω)
25		cardennn 9897	. . . . . . . . 9 ⊢ ((𝐴 ≈ suc 𝑚 ∧ suc 𝑚 ∈ ω) → (card‘𝐴) = suc 𝑚)
26	22, 24, 25	syl2anc 585	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → (card‘𝐴) = suc 𝑚)
27		cardennn 9897	. . . . . . . . . . 11 ⊢ (((𝐴 ∖ {𝑋}) ≈ 𝑚 ∧ 𝑚 ∈ ω) → (card‘(𝐴 ∖ {𝑋})) = 𝑚)
28	27	ancoms 458	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → (card‘(𝐴 ∖ {𝑋})) = 𝑚)
29	28	3adant1 1131	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → (card‘(𝐴 ∖ {𝑋})) = 𝑚)
30		suceq 6384	. . . . . . . . 9 ⊢ ((card‘(𝐴 ∖ {𝑋})) = 𝑚 → suc (card‘(𝐴 ∖ {𝑋})) = suc 𝑚)
31	29, 30	syl 17	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → suc (card‘(𝐴 ∖ {𝑋})) = suc 𝑚)
32	26, 31	eqtr4d 2773	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → (card‘𝐴) = suc (card‘(𝐴 ∖ {𝑋})))
33	32	3expib 1123	. . . . . 6 ⊢ (𝑋 ∈ 𝐴 → ((𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → (card‘𝐴) = suc (card‘(𝐴 ∖ {𝑋}))))
34	33	com12 32	. . . . 5 ⊢ ((𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → (𝑋 ∈ 𝐴 → (card‘𝐴) = suc (card‘(𝐴 ∖ {𝑋}))))
35	34	rexlimiva 3128	. . . 4 ⊢ (∃𝑚 ∈ ω (𝐴 ∖ {𝑋}) ≈ 𝑚 → (𝑋 ∈ 𝐴 → (card‘𝐴) = suc (card‘(𝐴 ∖ {𝑋}))))
36	2, 35	sylbi 217	. . 3 ⊢ ((𝐴 ∖ {𝑋}) ∈ Fin → (𝑋 ∈ 𝐴 → (card‘𝐴) = suc (card‘(𝐴 ∖ {𝑋}))))
37	1, 36	syl 17	. 2 ⊢ (𝐴 ∈ Fin → (𝑋 ∈ 𝐴 → (card‘𝐴) = suc (card‘(𝐴 ∖ {𝑋}))))
38	37	imp 406	1 ⊢ ((𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐴) → (card‘𝐴) = suc (card‘(𝐴 ∖ {𝑋})))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∃wrex 3059   ∖ cdif 3897   ∪ cun 3898   ∩ cin 3899  ∅c0 4284  {csn 4579   class class class wbr 5097  Ord word 6315  suc csuc 6318  ‘cfv 6491  ωcom 7808   ≈ cen 8882  Fincfn 8885  cardccrd 9849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4902  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-om 7809  df-1o 8397  df-er 8635  df-en 8886  df-dom 8887  df-sdom 8888  df-fin 8889  df-card 9853

This theorem is referenced by:  unidifsnel  32590  unidifsnne  32591