Theorem dif1en 7061

Description: If a set 𝐴 is equinumerous to the successor of a natural number 𝑀, then 𝐴 with an element removed is equinumerous to 𝑀. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 16-Aug-2015.)

Assertion

Ref	Expression
dif1en	⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)

Proof of Theorem dif1en

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1022	. . . 4 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → 𝐴 ≈ suc 𝑀)
2	1	ensymd 6952	. . 3 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → suc 𝑀 ≈ 𝐴)
3		bren 6912	. . 3 ⊢ (suc 𝑀 ≈ 𝐴 ↔ ∃𝑓 𝑓:suc 𝑀–1-1-onto→𝐴)
4	2, 3	sylib 122	. 2 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → ∃𝑓 𝑓:suc 𝑀–1-1-onto→𝐴)
5		peano2 4691	. . . . . . . 8 ⊢ (𝑀 ∈ ω → suc 𝑀 ∈ ω)
6		nnfi 7054	. . . . . . . 8 ⊢ (suc 𝑀 ∈ ω → suc 𝑀 ∈ Fin)
7	5, 6	syl 14	. . . . . . 7 ⊢ (𝑀 ∈ ω → suc 𝑀 ∈ Fin)
8	7	3ad2ant1 1042	. . . . . 6 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → suc 𝑀 ∈ Fin)
9		enfii 7056	. . . . . 6 ⊢ ((suc 𝑀 ∈ Fin ∧ 𝐴 ≈ suc 𝑀) → 𝐴 ∈ Fin)
10	8, 1, 9	syl2anc 411	. . . . 5 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → 𝐴 ∈ Fin)
11	10	adantr 276	. . . 4 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → 𝐴 ∈ Fin)
12		simpl3 1026	. . . 4 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → 𝑋 ∈ 𝐴)
13		f1of 5580	. . . . . 6 ⊢ (𝑓:suc 𝑀–1-1-onto→𝐴 → 𝑓:suc 𝑀⟶𝐴)
14	13	adantl 277	. . . . 5 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → 𝑓:suc 𝑀⟶𝐴)
15		sucidg 4511	. . . . . . 7 ⊢ (𝑀 ∈ ω → 𝑀 ∈ suc 𝑀)
16	15	3ad2ant1 1042	. . . . . 6 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → 𝑀 ∈ suc 𝑀)
17	16	adantr 276	. . . . 5 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → 𝑀 ∈ suc 𝑀)
18	14, 17	ffvelcdmd 5779	. . . 4 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (𝑓‘𝑀) ∈ 𝐴)
19		fidifsnen 7052	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐴 ∧ (𝑓‘𝑀) ∈ 𝐴) → (𝐴 ∖ {𝑋}) ≈ (𝐴 ∖ {(𝑓‘𝑀)}))
20	11, 12, 18, 19	syl3anc 1271	. . 3 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (𝐴 ∖ {𝑋}) ≈ (𝐴 ∖ {(𝑓‘𝑀)}))
21		nnord 4708	. . . . . . . 8 ⊢ (𝑀 ∈ ω → Ord 𝑀)
22		orddif 4643	. . . . . . . 8 ⊢ (Ord 𝑀 → 𝑀 = (suc 𝑀 ∖ {𝑀}))
23	21, 22	syl 14	. . . . . . 7 ⊢ (𝑀 ∈ ω → 𝑀 = (suc 𝑀 ∖ {𝑀}))
24	23	3ad2ant1 1042	. . . . . 6 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → 𝑀 = (suc 𝑀 ∖ {𝑀}))
25	24	adantr 276	. . . . 5 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → 𝑀 = (suc 𝑀 ∖ {𝑀}))
26	23	eleq1d 2298	. . . . . . . . 9 ⊢ (𝑀 ∈ ω → (𝑀 ∈ ω ↔ (suc 𝑀 ∖ {𝑀}) ∈ ω))
27	26	ibi 176	. . . . . . . 8 ⊢ (𝑀 ∈ ω → (suc 𝑀 ∖ {𝑀}) ∈ ω)
28	27	3ad2ant1 1042	. . . . . . 7 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (suc 𝑀 ∖ {𝑀}) ∈ ω)
29	28	adantr 276	. . . . . 6 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (suc 𝑀 ∖ {𝑀}) ∈ ω)
30		dff1o2 5585	. . . . . . . . 9 ⊢ (𝑓:suc 𝑀–1-1-onto→𝐴 ↔ (𝑓 Fn suc 𝑀 ∧ Fun ◡𝑓 ∧ ran 𝑓 = 𝐴))
31	30	simp2bi 1037	. . . . . . . 8 ⊢ (𝑓:suc 𝑀–1-1-onto→𝐴 → Fun ◡𝑓)
32	31	adantl 277	. . . . . . 7 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → Fun ◡𝑓)
33		f1ofo 5587	. . . . . . . . 9 ⊢ (𝑓:suc 𝑀–1-1-onto→𝐴 → 𝑓:suc 𝑀–onto→𝐴)
34	33	adantl 277	. . . . . . . 8 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → 𝑓:suc 𝑀–onto→𝐴)
35		f1orel 5583	. . . . . . . . . . . 12 ⊢ (𝑓:suc 𝑀–1-1-onto→𝐴 → Rel 𝑓)
36	35	adantl 277	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → Rel 𝑓)
37		resdm 5050	. . . . . . . . . . 11 ⊢ (Rel 𝑓 → (𝑓 ↾ dom 𝑓) = 𝑓)
38	36, 37	syl 14	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (𝑓 ↾ dom 𝑓) = 𝑓)
39		f1odm 5584	. . . . . . . . . . . 12 ⊢ (𝑓:suc 𝑀–1-1-onto→𝐴 → dom 𝑓 = suc 𝑀)
40	39	reseq2d 5011	. . . . . . . . . . 11 ⊢ (𝑓:suc 𝑀–1-1-onto→𝐴 → (𝑓 ↾ dom 𝑓) = (𝑓 ↾ suc 𝑀))
41	40	adantl 277	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (𝑓 ↾ dom 𝑓) = (𝑓 ↾ suc 𝑀))
42	38, 41	eqtr3d 2264	. . . . . . . . 9 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → 𝑓 = (𝑓 ↾ suc 𝑀))
43		foeq1 5552	. . . . . . . . 9 ⊢ (𝑓 = (𝑓 ↾ suc 𝑀) → (𝑓:suc 𝑀–onto→𝐴 ↔ (𝑓 ↾ suc 𝑀):suc 𝑀–onto→𝐴))
44	42, 43	syl 14	. . . . . . . 8 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (𝑓:suc 𝑀–onto→𝐴 ↔ (𝑓 ↾ suc 𝑀):suc 𝑀–onto→𝐴))
45	34, 44	mpbid 147	. . . . . . 7 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (𝑓 ↾ suc 𝑀):suc 𝑀–onto→𝐴)
46		simpl1 1024	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → 𝑀 ∈ ω)
47		f1osng 5622	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ω ∧ (𝑓‘𝑀) ∈ 𝐴) → {⟨𝑀, (𝑓‘𝑀)⟩}:{𝑀}–1-1-onto→{(𝑓‘𝑀)})
48	46, 18, 47	syl2anc 411	. . . . . . . . 9 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → {⟨𝑀, (𝑓‘𝑀)⟩}:{𝑀}–1-1-onto→{(𝑓‘𝑀)})
49		f1ofo 5587	. . . . . . . . 9 ⊢ ({⟨𝑀, (𝑓‘𝑀)⟩}:{𝑀}–1-1-onto→{(𝑓‘𝑀)} → {⟨𝑀, (𝑓‘𝑀)⟩}:{𝑀}–onto→{(𝑓‘𝑀)})
50	48, 49	syl 14	. . . . . . . 8 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → {⟨𝑀, (𝑓‘𝑀)⟩}:{𝑀}–onto→{(𝑓‘𝑀)})
51		f1ofn 5581	. . . . . . . . . . 11 ⊢ (𝑓:suc 𝑀–1-1-onto→𝐴 → 𝑓 Fn suc 𝑀)
52	51	adantl 277	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → 𝑓 Fn suc 𝑀)
53		fnressn 5835	. . . . . . . . . 10 ⊢ ((𝑓 Fn suc 𝑀 ∧ 𝑀 ∈ suc 𝑀) → (𝑓 ↾ {𝑀}) = {⟨𝑀, (𝑓‘𝑀)⟩})
54	52, 17, 53	syl2anc 411	. . . . . . . . 9 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (𝑓 ↾ {𝑀}) = {⟨𝑀, (𝑓‘𝑀)⟩})
55		foeq1 5552	. . . . . . . . 9 ⊢ ((𝑓 ↾ {𝑀}) = {⟨𝑀, (𝑓‘𝑀)⟩} → ((𝑓 ↾ {𝑀}):{𝑀}–onto→{(𝑓‘𝑀)} ↔ {⟨𝑀, (𝑓‘𝑀)⟩}:{𝑀}–onto→{(𝑓‘𝑀)}))
56	54, 55	syl 14	. . . . . . . 8 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → ((𝑓 ↾ {𝑀}):{𝑀}–onto→{(𝑓‘𝑀)} ↔ {⟨𝑀, (𝑓‘𝑀)⟩}:{𝑀}–onto→{(𝑓‘𝑀)}))
57	50, 56	mpbird 167	. . . . . . 7 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (𝑓 ↾ {𝑀}):{𝑀}–onto→{(𝑓‘𝑀)})
58		resdif 5602	. . . . . . 7 ⊢ ((Fun ◡𝑓 ∧ (𝑓 ↾ suc 𝑀):suc 𝑀–onto→𝐴 ∧ (𝑓 ↾ {𝑀}):{𝑀}–onto→{(𝑓‘𝑀)}) → (𝑓 ↾ (suc 𝑀 ∖ {𝑀})):(suc 𝑀 ∖ {𝑀})–1-1-onto→(𝐴 ∖ {(𝑓‘𝑀)}))
59	32, 45, 57, 58	syl3anc 1271	. . . . . 6 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (𝑓 ↾ (suc 𝑀 ∖ {𝑀})):(suc 𝑀 ∖ {𝑀})–1-1-onto→(𝐴 ∖ {(𝑓‘𝑀)}))
60		f1oeng 6925	. . . . . 6 ⊢ (((suc 𝑀 ∖ {𝑀}) ∈ ω ∧ (𝑓 ↾ (suc 𝑀 ∖ {𝑀})):(suc 𝑀 ∖ {𝑀})–1-1-onto→(𝐴 ∖ {(𝑓‘𝑀)})) → (suc 𝑀 ∖ {𝑀}) ≈ (𝐴 ∖ {(𝑓‘𝑀)}))
61	29, 59, 60	syl2anc 411	. . . . 5 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (suc 𝑀 ∖ {𝑀}) ≈ (𝐴 ∖ {(𝑓‘𝑀)}))
62	25, 61	eqbrtrd 4108	. . . 4 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → 𝑀 ≈ (𝐴 ∖ {(𝑓‘𝑀)}))
63	62	ensymd 6952	. . 3 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (𝐴 ∖ {(𝑓‘𝑀)}) ≈ 𝑀)
64		entr 6953	. . 3 ⊢ (((𝐴 ∖ {𝑋}) ≈ (𝐴 ∖ {(𝑓‘𝑀)}) ∧ (𝐴 ∖ {(𝑓‘𝑀)}) ≈ 𝑀) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
65	20, 63, 64	syl2anc 411	. 2 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
66	4, 65	exlimddv 1945	1 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1002   = wceq 1395  ∃wex 1538   ∈ wcel 2200   ∖ cdif 3195  {csn 3667  ⟨cop 3670   class class class wbr 4086  Ord word 4457  suc csuc 4460  ωcom 4686  ^◡ccnv 4722  dom cdm 4723  ran crn 4724   ↾ cres 4725  Rel wrel 4728  Fun wfun 5318   Fn wfn 5319  ⟶wf 5320  –onto→wfo 5322  –1-1-onto→wf1o 5323  ‘cfv 5324   ≈ cen 6902  Fincfn 6904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-er 6697  df-en 6905  df-fin 6907

This theorem is referenced by:  dif1enen  7062  findcard  7070  findcard2  7071  findcard2s  7072  diffisn  7075  en2eleq  7396  en2other2  7397  zfz1isolem1  11094