Theorem dif1card 9963

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 9139	. . 3 ⊢ (𝐴 ∈ Fin → (𝐴 ∖ {𝑋}) ∈ Fin)
2		isfi 8947	. . . 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 9012	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω) → {𝑋} ≈ {𝑚})
5	4	3adant3 1132	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → {𝑋} ≈ {𝑚})
6		disjdifr 4436	. . . . . . . . . . . 12 ⊢ ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅
7	6	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅)
8		nnord 7850	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ω → Ord 𝑚)
9		ordirr 6350	. . . . . . . . . . . . . 14 ⊢ (Ord 𝑚 → ¬ 𝑚 ∈ 𝑚)
10	8, 9	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ω → ¬ 𝑚 ∈ 𝑚)
11		disjsn 4675	. . . . . . . . . . . . 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 9017	. . . . . . . . . . 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 4774	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝐴 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴)
17		df-suc 6338	. . . . . . . . . . . . . 14 ⊢ suc 𝑚 = (𝑚 ∪ {𝑚})
18	17	eqcomi 2738	. . . . . . . . . . . . 13 ⊢ (𝑚 ∪ {𝑚}) = suc 𝑚
19	18	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝐴 → (𝑚 ∪ {𝑚}) = suc 𝑚)
20	16, 19	breq12d 5120	. . . . . . . . . . 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 7866	. . . . . . . . . 10 ⊢ (𝑚 ∈ ω → suc 𝑚 ∈ ω)
24	23	3ad2ant2 1134	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → suc 𝑚 ∈ ω)
25		cardennn 9936	. . . . . . . . 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 9936	. . . . . . . . . . 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 6400	. . . . . . . . 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 2767	. . . . . . 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 3126	. . . 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 1540   ∈ wcel 2109  ∃wrex 3053   ∖ cdif 3911   ∪ cun 3912   ∩ cin 3913  ∅c0 4296  {csn 4589   class class class wbr 5107  Ord word 6331  suc csuc 6334  ‘cfv 6511  ωcom 7842   ≈ cen 8915  Fincfn 8918  cardccrd 9888

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-om 7843  df-1o 8434  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-card 9892

This theorem is referenced by:  unidifsnel  32464  unidifsnne  32465