Theorem dif1card 9939

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 9116	. . 3 ⊢ (𝐴 ∈ Fin → (𝐴 ∖ {𝑋}) ∈ Fin)
2		isfi 8924	. . . 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 8989	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω) → {𝑋} ≈ {𝑚})
5	4	3adant3 1132	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → {𝑋} ≈ {𝑚})
6		disjdifr 4432	. . . . . . . . . . . 12 ⊢ ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅
7	6	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅)
8		nnord 7830	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ω → Ord 𝑚)
9		ordirr 6338	. . . . . . . . . . . . . 14 ⊢ (Ord 𝑚 → ¬ 𝑚 ∈ 𝑚)
10	8, 9	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ ω → ¬ 𝑚 ∈ 𝑚)
11		disjsn 4671	. . . . . . . . . . . . 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 8994	. . . . . . . . . . 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 4770	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝐴 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴)
17		df-suc 6326	. . . . . . . . . . . . . 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 5115	. . . . . . . . . . 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 7846	. . . . . . . . . 10 ⊢ (𝑚 ∈ ω → suc 𝑚 ∈ ω)
24	23	3ad2ant2 1134	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑚 ∈ ω ∧ (𝐴 ∖ {𝑋}) ≈ 𝑚) → suc 𝑚 ∈ ω)
25		cardennn 9912	. . . . . . . . 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 9912	. . . . . . . . . . 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 6388	. . . . . . . . 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 3908   ∪ cun 3909   ∩ cin 3910  ∅c0 4292  {csn 4585   class class class wbr 5102  Ord word 6319  suc csuc 6322  ‘cfv 6499  ωcom 7822   ≈ cen 8892  Fincfn 8895  cardccrd 9864

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 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

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 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-om 7823  df-1o 8411  df-er 8648  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-card 9868

This theorem is referenced by:  unidifsnel  32514  unidifsnne  32515