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Theorem infdifsn 8539
Description: Removing a singleton from an infinite set does not change the cardinality of the set. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Mario Carneiro, 16-May-2015.)
Assertion
Ref Expression
infdifsn (ω ≼ 𝐴 → (𝐴 ∖ {𝐵}) ≈ 𝐴)

Proof of Theorem infdifsn
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 brdomi 7951 . . . 4 (ω ≼ 𝐴 → ∃𝑓 𝑓:ω–1-1𝐴)
21adantr 481 . . 3 ((ω ≼ 𝐴𝐵𝐴) → ∃𝑓 𝑓:ω–1-1𝐴)
3 reldom 7946 . . . . . . 7 Rel ≼
43brrelex2i 5149 . . . . . 6 (ω ≼ 𝐴𝐴 ∈ V)
54ad2antrr 761 . . . . 5 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → 𝐴 ∈ V)
6 simplr 791 . . . . 5 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → 𝐵𝐴)
7 f1f 6088 . . . . . . 7 (𝑓:ω–1-1𝐴𝑓:ω⟶𝐴)
87adantl 482 . . . . . 6 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → 𝑓:ω⟶𝐴)
9 peano1 7070 . . . . . 6 ∅ ∈ ω
10 ffvelrn 6343 . . . . . 6 ((𝑓:ω⟶𝐴 ∧ ∅ ∈ ω) → (𝑓‘∅) ∈ 𝐴)
118, 9, 10sylancl 693 . . . . 5 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝑓‘∅) ∈ 𝐴)
12 difsnen 8027 . . . . 5 ((𝐴 ∈ V ∧ 𝐵𝐴 ∧ (𝑓‘∅) ∈ 𝐴) → (𝐴 ∖ {𝐵}) ≈ (𝐴 ∖ {(𝑓‘∅)}))
135, 6, 11, 12syl3anc 1324 . . . 4 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝐴 ∖ {𝐵}) ≈ (𝐴 ∖ {(𝑓‘∅)}))
14 vex 3198 . . . . . . . . . 10 𝑓 ∈ V
15 f1f1orn 6135 . . . . . . . . . . 11 (𝑓:ω–1-1𝐴𝑓:ω–1-1-onto→ran 𝑓)
1615adantl 482 . . . . . . . . . 10 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → 𝑓:ω–1-1-onto→ran 𝑓)
17 f1oen3g 7956 . . . . . . . . . 10 ((𝑓 ∈ V ∧ 𝑓:ω–1-1-onto→ran 𝑓) → ω ≈ ran 𝑓)
1814, 16, 17sylancr 694 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ω ≈ ran 𝑓)
1918ensymd 7992 . . . . . . . 8 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ran 𝑓 ≈ ω)
203brrelexi 5148 . . . . . . . . . . 11 (ω ≼ 𝐴 → ω ∈ V)
2120ad2antrr 761 . . . . . . . . . 10 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ω ∈ V)
22 limom 7065 . . . . . . . . . . 11 Lim ω
2322limenpsi 8120 . . . . . . . . . 10 (ω ∈ V → ω ≈ (ω ∖ {∅}))
2421, 23syl 17 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ω ≈ (ω ∖ {∅}))
2514resex 5431 . . . . . . . . . . 11 (𝑓 ↾ (ω ∖ {∅})) ∈ V
26 simpr 477 . . . . . . . . . . . 12 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → 𝑓:ω–1-1𝐴)
27 difss 3729 . . . . . . . . . . . 12 (ω ∖ {∅}) ⊆ ω
28 f1ores 6138 . . . . . . . . . . . 12 ((𝑓:ω–1-1𝐴 ∧ (ω ∖ {∅}) ⊆ ω) → (𝑓 ↾ (ω ∖ {∅})):(ω ∖ {∅})–1-1-onto→(𝑓 “ (ω ∖ {∅})))
2926, 27, 28sylancl 693 . . . . . . . . . . 11 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝑓 ↾ (ω ∖ {∅})):(ω ∖ {∅})–1-1-onto→(𝑓 “ (ω ∖ {∅})))
30 f1oen3g 7956 . . . . . . . . . . 11 (((𝑓 ↾ (ω ∖ {∅})) ∈ V ∧ (𝑓 ↾ (ω ∖ {∅})):(ω ∖ {∅})–1-1-onto→(𝑓 “ (ω ∖ {∅}))) → (ω ∖ {∅}) ≈ (𝑓 “ (ω ∖ {∅})))
3125, 29, 30sylancr 694 . . . . . . . . . 10 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (ω ∖ {∅}) ≈ (𝑓 “ (ω ∖ {∅})))
32 f1orn 6134 . . . . . . . . . . . . 13 (𝑓:ω–1-1-onto→ran 𝑓 ↔ (𝑓 Fn ω ∧ Fun 𝑓))
3332simprbi 480 . . . . . . . . . . . 12 (𝑓:ω–1-1-onto→ran 𝑓 → Fun 𝑓)
34 imadif 5961 . . . . . . . . . . . 12 (Fun 𝑓 → (𝑓 “ (ω ∖ {∅})) = ((𝑓 “ ω) ∖ (𝑓 “ {∅})))
3516, 33, 343syl 18 . . . . . . . . . . 11 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝑓 “ (ω ∖ {∅})) = ((𝑓 “ ω) ∖ (𝑓 “ {∅})))
36 f1fn 6089 . . . . . . . . . . . . . 14 (𝑓:ω–1-1𝐴𝑓 Fn ω)
3736adantl 482 . . . . . . . . . . . . 13 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → 𝑓 Fn ω)
38 fnima 5997 . . . . . . . . . . . . 13 (𝑓 Fn ω → (𝑓 “ ω) = ran 𝑓)
3937, 38syl 17 . . . . . . . . . . . 12 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝑓 “ ω) = ran 𝑓)
40 fnsnfv 6245 . . . . . . . . . . . . . 14 ((𝑓 Fn ω ∧ ∅ ∈ ω) → {(𝑓‘∅)} = (𝑓 “ {∅}))
4137, 9, 40sylancl 693 . . . . . . . . . . . . 13 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → {(𝑓‘∅)} = (𝑓 “ {∅}))
4241eqcomd 2626 . . . . . . . . . . . 12 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝑓 “ {∅}) = {(𝑓‘∅)})
4339, 42difeq12d 3721 . . . . . . . . . . 11 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ((𝑓 “ ω) ∖ (𝑓 “ {∅})) = (ran 𝑓 ∖ {(𝑓‘∅)}))
4435, 43eqtrd 2654 . . . . . . . . . 10 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝑓 “ (ω ∖ {∅})) = (ran 𝑓 ∖ {(𝑓‘∅)}))
4531, 44breqtrd 4670 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (ω ∖ {∅}) ≈ (ran 𝑓 ∖ {(𝑓‘∅)}))
46 entr 7993 . . . . . . . . 9 ((ω ≈ (ω ∖ {∅}) ∧ (ω ∖ {∅}) ≈ (ran 𝑓 ∖ {(𝑓‘∅)})) → ω ≈ (ran 𝑓 ∖ {(𝑓‘∅)}))
4724, 45, 46syl2anc 692 . . . . . . . 8 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ω ≈ (ran 𝑓 ∖ {(𝑓‘∅)}))
48 entr 7993 . . . . . . . 8 ((ran 𝑓 ≈ ω ∧ ω ≈ (ran 𝑓 ∖ {(𝑓‘∅)})) → ran 𝑓 ≈ (ran 𝑓 ∖ {(𝑓‘∅)}))
4919, 47, 48syl2anc 692 . . . . . . 7 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ran 𝑓 ≈ (ran 𝑓 ∖ {(𝑓‘∅)}))
50 difexg 4799 . . . . . . . 8 (𝐴 ∈ V → (𝐴 ∖ ran 𝑓) ∈ V)
51 enrefg 7972 . . . . . . . 8 ((𝐴 ∖ ran 𝑓) ∈ V → (𝐴 ∖ ran 𝑓) ≈ (𝐴 ∖ ran 𝑓))
525, 50, 513syl 18 . . . . . . 7 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝐴 ∖ ran 𝑓) ≈ (𝐴 ∖ ran 𝑓))
53 disjdif 4031 . . . . . . . 8 (ran 𝑓 ∩ (𝐴 ∖ ran 𝑓)) = ∅
5453a1i 11 . . . . . . 7 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (ran 𝑓 ∩ (𝐴 ∖ ran 𝑓)) = ∅)
55 difss 3729 . . . . . . . . . 10 (ran 𝑓 ∖ {(𝑓‘∅)}) ⊆ ran 𝑓
56 ssrin 3830 . . . . . . . . . 10 ((ran 𝑓 ∖ {(𝑓‘∅)}) ⊆ ran 𝑓 → ((ran 𝑓 ∖ {(𝑓‘∅)}) ∩ (𝐴 ∖ ran 𝑓)) ⊆ (ran 𝑓 ∩ (𝐴 ∖ ran 𝑓)))
5755, 56ax-mp 5 . . . . . . . . 9 ((ran 𝑓 ∖ {(𝑓‘∅)}) ∩ (𝐴 ∖ ran 𝑓)) ⊆ (ran 𝑓 ∩ (𝐴 ∖ ran 𝑓))
58 sseq0 3966 . . . . . . . . 9 ((((ran 𝑓 ∖ {(𝑓‘∅)}) ∩ (𝐴 ∖ ran 𝑓)) ⊆ (ran 𝑓 ∩ (𝐴 ∖ ran 𝑓)) ∧ (ran 𝑓 ∩ (𝐴 ∖ ran 𝑓)) = ∅) → ((ran 𝑓 ∖ {(𝑓‘∅)}) ∩ (𝐴 ∖ ran 𝑓)) = ∅)
5957, 53, 58mp2an 707 . . . . . . . 8 ((ran 𝑓 ∖ {(𝑓‘∅)}) ∩ (𝐴 ∖ ran 𝑓)) = ∅
6059a1i 11 . . . . . . 7 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ((ran 𝑓 ∖ {(𝑓‘∅)}) ∩ (𝐴 ∖ ran 𝑓)) = ∅)
61 unen 8025 . . . . . . 7 (((ran 𝑓 ≈ (ran 𝑓 ∖ {(𝑓‘∅)}) ∧ (𝐴 ∖ ran 𝑓) ≈ (𝐴 ∖ ran 𝑓)) ∧ ((ran 𝑓 ∩ (𝐴 ∖ ran 𝑓)) = ∅ ∧ ((ran 𝑓 ∖ {(𝑓‘∅)}) ∩ (𝐴 ∖ ran 𝑓)) = ∅)) → (ran 𝑓 ∪ (𝐴 ∖ ran 𝑓)) ≈ ((ran 𝑓 ∖ {(𝑓‘∅)}) ∪ (𝐴 ∖ ran 𝑓)))
6249, 52, 54, 60, 61syl22anc 1325 . . . . . 6 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (ran 𝑓 ∪ (𝐴 ∖ ran 𝑓)) ≈ ((ran 𝑓 ∖ {(𝑓‘∅)}) ∪ (𝐴 ∖ ran 𝑓)))
63 frn 6040 . . . . . . . 8 (𝑓:ω⟶𝐴 → ran 𝑓𝐴)
648, 63syl 17 . . . . . . 7 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ran 𝑓𝐴)
65 undif 4040 . . . . . . 7 (ran 𝑓𝐴 ↔ (ran 𝑓 ∪ (𝐴 ∖ ran 𝑓)) = 𝐴)
6664, 65sylib 208 . . . . . 6 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (ran 𝑓 ∪ (𝐴 ∖ ran 𝑓)) = 𝐴)
67 uncom 3749 . . . . . . 7 ((ran 𝑓 ∖ {(𝑓‘∅)}) ∪ (𝐴 ∖ ran 𝑓)) = ((𝐴 ∖ ran 𝑓) ∪ (ran 𝑓 ∖ {(𝑓‘∅)}))
68 eldifn 3725 . . . . . . . . . . 11 ((𝑓‘∅) ∈ (𝐴 ∖ ran 𝑓) → ¬ (𝑓‘∅) ∈ ran 𝑓)
69 fnfvelrn 6342 . . . . . . . . . . . 12 ((𝑓 Fn ω ∧ ∅ ∈ ω) → (𝑓‘∅) ∈ ran 𝑓)
7037, 9, 69sylancl 693 . . . . . . . . . . 11 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝑓‘∅) ∈ ran 𝑓)
7168, 70nsyl3 133 . . . . . . . . . 10 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ¬ (𝑓‘∅) ∈ (𝐴 ∖ ran 𝑓))
72 disjsn 4237 . . . . . . . . . 10 (((𝐴 ∖ ran 𝑓) ∩ {(𝑓‘∅)}) = ∅ ↔ ¬ (𝑓‘∅) ∈ (𝐴 ∖ ran 𝑓))
7371, 72sylibr 224 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ((𝐴 ∖ ran 𝑓) ∩ {(𝑓‘∅)}) = ∅)
74 undif4 4026 . . . . . . . . 9 (((𝐴 ∖ ran 𝑓) ∩ {(𝑓‘∅)}) = ∅ → ((𝐴 ∖ ran 𝑓) ∪ (ran 𝑓 ∖ {(𝑓‘∅)})) = (((𝐴 ∖ ran 𝑓) ∪ ran 𝑓) ∖ {(𝑓‘∅)}))
7573, 74syl 17 . . . . . . . 8 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ((𝐴 ∖ ran 𝑓) ∪ (ran 𝑓 ∖ {(𝑓‘∅)})) = (((𝐴 ∖ ran 𝑓) ∪ ran 𝑓) ∖ {(𝑓‘∅)}))
76 uncom 3749 . . . . . . . . . 10 ((𝐴 ∖ ran 𝑓) ∪ ran 𝑓) = (ran 𝑓 ∪ (𝐴 ∖ ran 𝑓))
7776, 66syl5eq 2666 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ((𝐴 ∖ ran 𝑓) ∪ ran 𝑓) = 𝐴)
7877difeq1d 3719 . . . . . . . 8 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (((𝐴 ∖ ran 𝑓) ∪ ran 𝑓) ∖ {(𝑓‘∅)}) = (𝐴 ∖ {(𝑓‘∅)}))
7975, 78eqtrd 2654 . . . . . . 7 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ((𝐴 ∖ ran 𝑓) ∪ (ran 𝑓 ∖ {(𝑓‘∅)})) = (𝐴 ∖ {(𝑓‘∅)}))
8067, 79syl5eq 2666 . . . . . 6 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ((ran 𝑓 ∖ {(𝑓‘∅)}) ∪ (𝐴 ∖ ran 𝑓)) = (𝐴 ∖ {(𝑓‘∅)}))
8162, 66, 803brtr3d 4675 . . . . 5 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → 𝐴 ≈ (𝐴 ∖ {(𝑓‘∅)}))
8281ensymd 7992 . . . 4 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝐴 ∖ {(𝑓‘∅)}) ≈ 𝐴)
83 entr 7993 . . . 4 (((𝐴 ∖ {𝐵}) ≈ (𝐴 ∖ {(𝑓‘∅)}) ∧ (𝐴 ∖ {(𝑓‘∅)}) ≈ 𝐴) → (𝐴 ∖ {𝐵}) ≈ 𝐴)
8413, 82, 83syl2anc 692 . . 3 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝐴 ∖ {𝐵}) ≈ 𝐴)
852, 84exlimddv 1861 . 2 ((ω ≼ 𝐴𝐵𝐴) → (𝐴 ∖ {𝐵}) ≈ 𝐴)
86 difsn 4319 . . . 4 𝐵𝐴 → (𝐴 ∖ {𝐵}) = 𝐴)
8786adantl 482 . . 3 ((ω ≼ 𝐴 ∧ ¬ 𝐵𝐴) → (𝐴 ∖ {𝐵}) = 𝐴)
88 enrefg 7972 . . . . 5 (𝐴 ∈ V → 𝐴𝐴)
894, 88syl 17 . . . 4 (ω ≼ 𝐴𝐴𝐴)
9089adantr 481 . . 3 ((ω ≼ 𝐴 ∧ ¬ 𝐵𝐴) → 𝐴𝐴)
9187, 90eqbrtrd 4666 . 2 ((ω ≼ 𝐴 ∧ ¬ 𝐵𝐴) → (𝐴 ∖ {𝐵}) ≈ 𝐴)
9285, 91pm2.61dan 831 1 (ω ≼ 𝐴 → (𝐴 ∖ {𝐵}) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1481  wex 1702  wcel 1988  Vcvv 3195  cdif 3564  cun 3565  cin 3566  wss 3567  c0 3907  {csn 4168   class class class wbr 4644  ccnv 5103  ran crn 5105  cres 5106  cima 5107  Fun wfun 5870   Fn wfn 5871  wf 5872  1-1wf1 5873  1-1-ontowf1o 5875  cfv 5876  ωcom 7050  cen 7937  cdom 7938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-om 7051  df-1o 7545  df-er 7727  df-en 7941  df-dom 7942
This theorem is referenced by:  infdiffi  8540  infcda1  9000  infpss  9024
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