Theorem dif1en 6991

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 1001	. . . 4 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → 𝐴 ≈ suc 𝑀)
2	1	ensymd 6888	. . 3 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → suc 𝑀 ≈ 𝐴)
3		bren 6848	. . 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 4651	. . . . . . . 8 ⊢ (𝑀 ∈ ω → suc 𝑀 ∈ ω)
6		nnfi 6984	. . . . . . . 8 ⊢ (suc 𝑀 ∈ ω → suc 𝑀 ∈ Fin)
7	5, 6	syl 14	. . . . . . 7 ⊢ (𝑀 ∈ ω → suc 𝑀 ∈ Fin)
8	7	3ad2ant1 1021	. . . . . 6 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → suc 𝑀 ∈ Fin)
9		enfii 6986	. . . . . 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 1005	. . . 4 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → 𝑋 ∈ 𝐴)
13		f1of 5534	. . . . . 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 4471	. . . . . . 7 ⊢ (𝑀 ∈ ω → 𝑀 ∈ suc 𝑀)
16	15	3ad2ant1 1021	. . . . . 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 5729	. . . 4 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (𝑓‘𝑀) ∈ 𝐴)
19		fidifsnen 6982	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝑋 ∈ 𝐴 ∧ (𝑓‘𝑀) ∈ 𝐴) → (𝐴 ∖ {𝑋}) ≈ (𝐴 ∖ {(𝑓‘𝑀)}))
20	11, 12, 18, 19	syl3anc 1250	. . 3 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (𝐴 ∖ {𝑋}) ≈ (𝐴 ∖ {(𝑓‘𝑀)}))
21		nnord 4668	. . . . . . . 8 ⊢ (𝑀 ∈ ω → Ord 𝑀)
22		orddif 4603	. . . . . . . 8 ⊢ (Ord 𝑀 → 𝑀 = (suc 𝑀 ∖ {𝑀}))
23	21, 22	syl 14	. . . . . . 7 ⊢ (𝑀 ∈ ω → 𝑀 = (suc 𝑀 ∖ {𝑀}))
24	23	3ad2ant1 1021	. . . . . 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 2275	. . . . . . . . 9 ⊢ (𝑀 ∈ ω → (𝑀 ∈ ω ↔ (suc 𝑀 ∖ {𝑀}) ∈ ω))
27	26	ibi 176	. . . . . . . 8 ⊢ (𝑀 ∈ ω → (suc 𝑀 ∖ {𝑀}) ∈ ω)
28	27	3ad2ant1 1021	. . . . . . 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 5539	. . . . . . . . 9 ⊢ (𝑓:suc 𝑀–1-1-onto→𝐴 ↔ (𝑓 Fn suc 𝑀 ∧ Fun ◡𝑓 ∧ ran 𝑓 = 𝐴))
31	30	simp2bi 1016	. . . . . . . 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 5541	. . . . . . . . 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 5537	. . . . . . . . . . . 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 5007	. . . . . . . . . . 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 5538	. . . . . . . . . . . 12 ⊢ (𝑓:suc 𝑀–1-1-onto→𝐴 → dom 𝑓 = suc 𝑀)
40	39	reseq2d 4968	. . . . . . . . . . 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 2241	. . . . . . . . 9 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → 𝑓 = (𝑓 ↾ suc 𝑀))
43		foeq1 5506	. . . . . . . . 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 1003	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → 𝑀 ∈ ω)
47		f1osng 5576	. . . . . . . . . 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 5541	. . . . . . . . 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 5535	. . . . . . . . . . 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 5783	. . . . . . . . . 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 5506	. . . . . . . . 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 5556	. . . . . . 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 1250	. . . . . 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 6861	. . . . . 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 4073	. . . 4 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → 𝑀 ≈ (𝐴 ∖ {(𝑓‘𝑀)}))
63	62	ensymd 6888	. . 3 ⊢ (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) ∧ 𝑓:suc 𝑀–1-1-onto→𝐴) → (𝐴 ∖ {(𝑓‘𝑀)}) ≈ 𝑀)
64		entr 6889	. . 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 1923	1 ⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 981   = wceq 1373  ∃wex 1516   ∈ wcel 2177   ∖ cdif 3167  {csn 3638  ⟨cop 3641   class class class wbr 4051  Ord word 4417  suc csuc 4420  ωcom 4646  ^◡ccnv 4682  dom cdm 4683  ran crn 4684   ↾ cres 4685  Rel wrel 4688  Fun wfun 5274   Fn wfn 5275  ⟶wf 5276  –onto→wfo 5278  –1-1-onto→wf1o 5279  ‘cfv 5280   ≈ cen 6838  Fincfn 6840

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-iinf 4644

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-tr 4151  df-id 4348  df-iord 4421  df-on 4423  df-suc 4426  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-er 6633  df-en 6841  df-fin 6843

This theorem is referenced by:  dif1enen  6992  findcard  7000  findcard2  7001  findcard2s  7002  diffisn  7005  en2eleq  7319  en2other2  7320  zfz1isolem1  11007