Theorem dif1en 7138

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 1025	. . . 4 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → 𝐴 ≈ suc 𝑀)
2	1	ensymd 7025	. . 3 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → suc 𝑀 ≈ 𝐴)
3		bren 6985	. . 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 4719	. . . . . . . 8 ⊢ (𝑀 ∈ ω → suc 𝑀 ∈ ω)
6		nnfi 7129	. . . . . . . 8 ⊢ (suc 𝑀 ∈ ω → suc 𝑀 ∈ Fin)
7	5, 6	syl 14	. . . . . . 7 ⊢ (𝑀 ∈ ω → suc 𝑀 ∈ Fin)
8	7	3ad2ant1 1045	. . . . . 6 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → suc 𝑀 ∈ Fin)
9		enfii 7131	. . . . . 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 1029	. . . 4 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → 𝑋 ∈ 𝐴)
13		f1of 5616	. . . . . 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 4539	. . . . . . 7 ⊢ (𝑀 ∈ ω → 𝑀 ∈ suc 𝑀)
16	15	3ad2ant1 1045	. . . . . 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 5815	. . . 4 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (𝑓‘𝑀) ∈ 𝐴)
19		fidifsnen 7127	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐴 ∧ (𝑓‘𝑀) ∈ 𝐴) → (𝐴 ∖ {𝑋}) ≈ (𝐴 ∖ {(𝑓‘𝑀)}))
20	11, 12, 18, 19	syl3anc 1274	. . 3 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (𝐴 ∖ {𝑋}) ≈ (𝐴 ∖ {(𝑓‘𝑀)}))
21		nnord 4736	. . . . . . . 8 ⊢ (𝑀 ∈ ω → Ord 𝑀)
22		orddif 4671	. . . . . . . 8 ⊢ (Ord 𝑀 → 𝑀 = (suc 𝑀 ∖ {𝑀}))
23	21, 22	syl 14	. . . . . . 7 ⊢ (𝑀 ∈ ω → 𝑀 = (suc 𝑀 ∖ {𝑀}))
24	23	3ad2ant1 1045	. . . . . 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 2303	. . . . . . . . 9 ⊢ (𝑀 ∈ ω → (𝑀 ∈ ω ↔ (suc 𝑀 ∖ {𝑀}) ∈ ω))
27	26	ibi 176	. . . . . . . 8 ⊢ (𝑀 ∈ ω → (suc 𝑀 ∖ {𝑀}) ∈ ω)
28	27	3ad2ant1 1045	. . . . . . 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 5621	. . . . . . . . 9 ⊢ (𝑓:suc 𝑀–1-1-onto→𝐴 ↔ (𝑓 Fn suc 𝑀 ∧ Fun ◡𝑓 ∧ ran 𝑓 = 𝐴))
31	30	simp2bi 1040	. . . . . . . 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 5623	. . . . . . . . 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 5619	. . . . . . . . . . . 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 5079	. . . . . . . . . . 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 5620	. . . . . . . . . . . 12 ⊢ (𝑓:suc 𝑀–1-1-onto→𝐴 → dom 𝑓 = suc 𝑀)
40	39	reseq2d 5040	. . . . . . . . . . 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 2269	. . . . . . . . 9 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → 𝑓 = (𝑓 ↾ suc 𝑀))
43		foeq1 5588	. . . . . . . . 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 1027	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → 𝑀 ∈ ω)
47		f1osng 5659	. . . . . . . . . 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 5623	. . . . . . . . 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 5617	. . . . . . . . . . 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 5872	. . . . . . . . . 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 5588	. . . . . . . . 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 5638	. . . . . . 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 1274	. . . . . 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 6998	. . . . . 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 4133	. . . 4 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → 𝑀 ≈ (𝐴 ∖ {(𝑓‘𝑀)}))
63	62	ensymd 7025	. . 3 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (𝐴 ∖ {(𝑓‘𝑀)}) ≈ 𝑀)
64		entr 7026	. . 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 1950	1 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1005   = wceq 1398  ∃wex 1541   ∈ wcel 2205   ∖ cdif 3210  {csn 3691  ⟨cop 3694   class class class wbr 4111  Ord word 4485  suc csuc 4488  ωcom 4714  ^◡ccnv 4750  dom cdm 4751  ran crn 4752   ↾ cres 4753  Rel wrel 4756  Fun wfun 5348   Fn wfn 5349  ⟶wf 5350  –onto→wfo 5352  –1-1-onto→wf1o 5353  ‘cfv 5354   ≈ cen 6975  Fincfn 6977

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-iord 4489  df-on 4491  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-er 6769  df-en 6978  df-fin 6980

This theorem is referenced by:  dif1enen  7139  findcard  7147  findcard2  7148  findcard2s  7149  diffisn  7152  en2eleq  7500  en2other2  7501  zfz1isolem1  11216