Theorem dif1card 10047

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 9213	. . 3 ⊢ (𝐴 ∈ Fin → (𝐴 ∖ {𝑋}) ∈ Fin)
2		isfi 9014	. . . 4 ⊢ ((𝐴 ∖ {𝑋}) ∈ Fin ↔ ∃𝑚 ∈ ω (𝐴 ∖ {𝑋}) ≈ 𝑚)
3		simp3 1137	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → (𝐴 ∖ {𝑋}) ≈ 𝑚)
4		en2sn 9079	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω) → {𝑋} ≈ {𝑚})
5	4	3adant3 1131	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → {𝑋} ≈ {𝑚})
6		disjdifr 4478	. . . . . . . . . . . 12 ⊢ ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅
7	6	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅)
8		nnord 7894	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ω → Ord 𝑚)
9		ordirr 6403	. . . . . . . . . . . . . 14 ⊢ (Ord 𝑚 → ¬ 𝑚 ∈ 𝑚)
10	8, 9	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ω → ¬ 𝑚 ∈ 𝑚)
11		disjsn 4715	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∩ {𝑚}) = ∅ ↔ ¬ 𝑚 ∈ 𝑚)
12	10, 11	sylibr 234	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ω → (𝑚 ∩ {𝑚}) = ∅)
13	12	3ad2ant2 1133	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → (𝑚 ∩ {𝑚}) = ∅)
14		unen 9084	. . . . . . . . . . 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 4814	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝐴 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴)
17		df-suc 6391	. . . . . . . . . . . . . 14 ⊢ suc 𝑚 = (𝑚 ∪ {𝑚})
18	17	eqcomi 2743	. . . . . . . . . . . . 13 ⊢ (𝑚 ∪ {𝑚}) = suc 𝑚
19	18	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝐴 → (𝑚 ∪ {𝑚}) = suc 𝑚)
20	16, 19	breq12d 5160	. . . . . . . . . . 11 ⊢ (𝑋 ∈ 𝐴 → (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑚 ∪ {𝑚}) ↔ 𝐴 ≈ suc 𝑚))
21	20	3ad2ant1 1132	. . . . . . . . . 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 7912	. . . . . . . . . 10 ⊢ (𝑚 ∈ ω → suc 𝑚 ∈ ω)
24	23	3ad2ant2 1133	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → suc 𝑚 ∈ ω)
25		cardennn 10020	. . . . . . . . 9 ⊢ ((𝐴 ≈ suc 𝑚 ∧ suc 𝑚 ∈ ω) → (card‘𝐴) = suc 𝑚)
26	22, 24, 25	syl2anc 584	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → (card‘𝐴) = suc 𝑚)
27		cardennn 10020	. . . . . . . . . . 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 1129	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → (card‘(𝐴 ∖ {𝑋})) = 𝑚)
30		suceq 6451	. . . . . . . . 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 2777	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → (card‘𝐴) = suc (card‘(𝐴 ∖ {𝑋})))
33	32	3expib 1121	. . . . . 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 3144	. . . 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 1086   = wceq 1536   ∈ wcel 2105  ∃wrex 3067   ∖ cdif 3959   ∪ cun 3960   ∩ cin 3961  ∅c0 4338  {csn 4630   class class class wbr 5147  Ord word 6384  suc csuc 6387  ‘cfv 6562  ωcom 7886   ≈ cen 8980  Fincfn 8983  cardccrd 9972

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-om 7887  df-1o 8504  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-card 9976

This theorem is referenced by:  unidifsnel  32560  unidifsnne  32561