Theorem dif1card 10079

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 9242	. . 3 ⊢ (𝐴 ∈ Fin → (𝐴 ∖ {𝑋}) ∈ Fin)
2		isfi 9036	. . . 4 ⊢ ((𝐴 ∖ {𝑋}) ∈ Fin ↔ ∃𝑚 ∈ ω (𝐴 ∖ {𝑋}) ≈ 𝑚)
3		simp3 1138	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → (𝐴 ∖ {𝑋}) ≈ 𝑚)
4		en2sn 9106	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω) → {𝑋} ≈ {𝑚})
5	4	3adant3 1132	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → {𝑋} ≈ {𝑚})
6		disjdifr 4496	. . . . . . . . . . . 12 ⊢ ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅
7	6	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅)
8		nnord 7911	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ω → Ord 𝑚)
9		ordirr 6413	. . . . . . . . . . . . . 14 ⊢ (Ord 𝑚 → ¬ 𝑚 ∈ 𝑚)
10	8, 9	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ω → ¬ 𝑚 ∈ 𝑚)
11		disjsn 4736	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∩ {𝑚}) = ∅ ↔ ¬ 𝑚 ∈ 𝑚)
12	10, 11	sylibr 234	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ω → (𝑚 ∩ {𝑚}) = ∅)
13	12	3ad2ant2 1134	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → (𝑚 ∩ {𝑚}) = ∅)
14		unen 9112	. . . . . . . . . . 11 ⊢ ((((𝐴 ∖ {𝑋}) ≈ 𝑚 ∧ {𝑋} ≈ {𝑚}) ∧ (((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅ ∧ (𝑚 ∩ {𝑚}) = ∅)) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑚 ∪ {𝑚}))
15	3, 5, 7, 13, 14	syl22anc 838	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑚 ∪ {𝑚}))
16		difsnid 4835	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝐴 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴)
17		df-suc 6401	. . . . . . . . . . . . . 14 ⊢ suc 𝑚 = (𝑚 ∪ {𝑚})
18	17	eqcomi 2749	. . . . . . . . . . . . 13 ⊢ (𝑚 ∪ {𝑚}) = suc 𝑚
19	18	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝐴 → (𝑚 ∪ {𝑚}) = suc 𝑚)
20	16, 19	breq12d 5179	. . . . . . . . . . 11 ⊢ (𝑋 ∈ 𝐴 → (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑚 ∪ {𝑚}) ↔ 𝐴 ≈ suc 𝑚))
21	20	3ad2ant1 1133	. . . . . . . . . 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 7929	. . . . . . . . . 10 ⊢ (𝑚 ∈ ω → suc 𝑚 ∈ ω)
24	23	3ad2ant2 1134	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → suc 𝑚 ∈ ω)
25		cardennn 10052	. . . . . . . . 9 ⊢ ((𝐴 ≈ suc 𝑚 ∧ suc 𝑚 ∈ ω) → (card‘𝐴) = suc 𝑚)
26	22, 24, 25	syl2anc 583	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → (card‘𝐴) = suc 𝑚)
27		cardennn 10052	. . . . . . . . . . 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 1130	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → (card‘(𝐴 ∖ {𝑋})) = 𝑚)
30		suceq 6461	. . . . . . . . 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 2783	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → (card‘𝐴) = suc (card‘(𝐴 ∖ {𝑋})))
33	32	3expib 1122	. . . . . 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 3153	. . . 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 1537   ∈ wcel 2108  ∃wrex 3076   ∖ cdif 3973   ∪ cun 3974   ∩ cin 3975  ∅c0 4352  {csn 4648   class class class wbr 5166  Ord word 6394  suc csuc 6397  ‘cfv 6573  ωcom 7903   ≈ cen 9000  Fincfn 9003  cardccrd 10004

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-om 7904  df-1o 8522  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-card 10008

This theorem is referenced by:  unidifsnel  32563  unidifsnne  32564