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Theorem sucdom2 8197
Description: Strict dominance of a set over another set implies dominance over its successor. (Contributed by Mario Carneiro, 12-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)
Assertion
Ref Expression
sucdom2 (𝐴𝐵 → suc 𝐴𝐵)

Proof of Theorem sucdom2
Dummy variables 𝑤 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sdomdom 8025 . . 3 (𝐴𝐵𝐴𝐵)
2 brdomi 8008 . . 3 (𝐴𝐵 → ∃𝑓 𝑓:𝐴1-1𝐵)
31, 2syl 17 . 2 (𝐴𝐵 → ∃𝑓 𝑓:𝐴1-1𝐵)
4 relsdom 8004 . . . . . . 7 Rel ≺
54brrelexi 5192 . . . . . 6 (𝐴𝐵𝐴 ∈ V)
65adantr 480 . . . . 5 ((𝐴𝐵𝑓:𝐴1-1𝐵) → 𝐴 ∈ V)
7 vex 3234 . . . . . . 7 𝑓 ∈ V
87rnex 7142 . . . . . 6 ran 𝑓 ∈ V
98a1i 11 . . . . 5 ((𝐴𝐵𝑓:𝐴1-1𝐵) → ran 𝑓 ∈ V)
10 f1f1orn 6186 . . . . . . 7 (𝑓:𝐴1-1𝐵𝑓:𝐴1-1-onto→ran 𝑓)
1110adantl 481 . . . . . 6 ((𝐴𝐵𝑓:𝐴1-1𝐵) → 𝑓:𝐴1-1-onto→ran 𝑓)
12 f1of1 6174 . . . . . 6 (𝑓:𝐴1-1-onto→ran 𝑓𝑓:𝐴1-1→ran 𝑓)
1311, 12syl 17 . . . . 5 ((𝐴𝐵𝑓:𝐴1-1𝐵) → 𝑓:𝐴1-1→ran 𝑓)
14 f1dom2g 8015 . . . . 5 ((𝐴 ∈ V ∧ ran 𝑓 ∈ V ∧ 𝑓:𝐴1-1→ran 𝑓) → 𝐴 ≼ ran 𝑓)
156, 9, 13, 14syl3anc 1366 . . . 4 ((𝐴𝐵𝑓:𝐴1-1𝐵) → 𝐴 ≼ ran 𝑓)
16 sdomnen 8026 . . . . . . . 8 (𝐴𝐵 → ¬ 𝐴𝐵)
1716adantr 480 . . . . . . 7 ((𝐴𝐵𝑓:𝐴1-1𝐵) → ¬ 𝐴𝐵)
18 ssdif0 3975 . . . . . . . 8 (𝐵 ⊆ ran 𝑓 ↔ (𝐵 ∖ ran 𝑓) = ∅)
19 simplr 807 . . . . . . . . . . 11 (((𝐴𝐵𝑓:𝐴1-1𝐵) ∧ 𝐵 ⊆ ran 𝑓) → 𝑓:𝐴1-1𝐵)
20 f1f 6139 . . . . . . . . . . . . . . 15 (𝑓:𝐴1-1𝐵𝑓:𝐴𝐵)
21 df-f 5930 . . . . . . . . . . . . . . 15 (𝑓:𝐴𝐵 ↔ (𝑓 Fn 𝐴 ∧ ran 𝑓𝐵))
2220, 21sylib 208 . . . . . . . . . . . . . 14 (𝑓:𝐴1-1𝐵 → (𝑓 Fn 𝐴 ∧ ran 𝑓𝐵))
2322simprd 478 . . . . . . . . . . . . 13 (𝑓:𝐴1-1𝐵 → ran 𝑓𝐵)
2419, 23syl 17 . . . . . . . . . . . 12 (((𝐴𝐵𝑓:𝐴1-1𝐵) ∧ 𝐵 ⊆ ran 𝑓) → ran 𝑓𝐵)
25 simpr 476 . . . . . . . . . . . 12 (((𝐴𝐵𝑓:𝐴1-1𝐵) ∧ 𝐵 ⊆ ran 𝑓) → 𝐵 ⊆ ran 𝑓)
2624, 25eqssd 3653 . . . . . . . . . . 11 (((𝐴𝐵𝑓:𝐴1-1𝐵) ∧ 𝐵 ⊆ ran 𝑓) → ran 𝑓 = 𝐵)
27 dff1o5 6184 . . . . . . . . . . 11 (𝑓:𝐴1-1-onto𝐵 ↔ (𝑓:𝐴1-1𝐵 ∧ ran 𝑓 = 𝐵))
2819, 26, 27sylanbrc 699 . . . . . . . . . 10 (((𝐴𝐵𝑓:𝐴1-1𝐵) ∧ 𝐵 ⊆ ran 𝑓) → 𝑓:𝐴1-1-onto𝐵)
29 f1oen3g 8013 . . . . . . . . . 10 ((𝑓 ∈ V ∧ 𝑓:𝐴1-1-onto𝐵) → 𝐴𝐵)
307, 28, 29sylancr 696 . . . . . . . . 9 (((𝐴𝐵𝑓:𝐴1-1𝐵) ∧ 𝐵 ⊆ ran 𝑓) → 𝐴𝐵)
3130ex 449 . . . . . . . 8 ((𝐴𝐵𝑓:𝐴1-1𝐵) → (𝐵 ⊆ ran 𝑓𝐴𝐵))
3218, 31syl5bir 233 . . . . . . 7 ((𝐴𝐵𝑓:𝐴1-1𝐵) → ((𝐵 ∖ ran 𝑓) = ∅ → 𝐴𝐵))
3317, 32mtod 189 . . . . . 6 ((𝐴𝐵𝑓:𝐴1-1𝐵) → ¬ (𝐵 ∖ ran 𝑓) = ∅)
34 neq0 3963 . . . . . 6 (¬ (𝐵 ∖ ran 𝑓) = ∅ ↔ ∃𝑤 𝑤 ∈ (𝐵 ∖ ran 𝑓))
3533, 34sylib 208 . . . . 5 ((𝐴𝐵𝑓:𝐴1-1𝐵) → ∃𝑤 𝑤 ∈ (𝐵 ∖ ran 𝑓))
36 snssi 4371 . . . . . . 7 (𝑤 ∈ (𝐵 ∖ ran 𝑓) → {𝑤} ⊆ (𝐵 ∖ ran 𝑓))
37 vex 3234 . . . . . . . . 9 𝑤 ∈ V
38 en2sn 8078 . . . . . . . . 9 ((𝐴 ∈ V ∧ 𝑤 ∈ V) → {𝐴} ≈ {𝑤})
396, 37, 38sylancl 695 . . . . . . . 8 ((𝐴𝐵𝑓:𝐴1-1𝐵) → {𝐴} ≈ {𝑤})
404brrelex2i 5193 . . . . . . . . . 10 (𝐴𝐵𝐵 ∈ V)
4140adantr 480 . . . . . . . . 9 ((𝐴𝐵𝑓:𝐴1-1𝐵) → 𝐵 ∈ V)
42 difexg 4841 . . . . . . . . 9 (𝐵 ∈ V → (𝐵 ∖ ran 𝑓) ∈ V)
43 ssdomg 8043 . . . . . . . . 9 ((𝐵 ∖ ran 𝑓) ∈ V → ({𝑤} ⊆ (𝐵 ∖ ran 𝑓) → {𝑤} ≼ (𝐵 ∖ ran 𝑓)))
4441, 42, 433syl 18 . . . . . . . 8 ((𝐴𝐵𝑓:𝐴1-1𝐵) → ({𝑤} ⊆ (𝐵 ∖ ran 𝑓) → {𝑤} ≼ (𝐵 ∖ ran 𝑓)))
45 endomtr 8055 . . . . . . . 8 (({𝐴} ≈ {𝑤} ∧ {𝑤} ≼ (𝐵 ∖ ran 𝑓)) → {𝐴} ≼ (𝐵 ∖ ran 𝑓))
4639, 44, 45syl6an 567 . . . . . . 7 ((𝐴𝐵𝑓:𝐴1-1𝐵) → ({𝑤} ⊆ (𝐵 ∖ ran 𝑓) → {𝐴} ≼ (𝐵 ∖ ran 𝑓)))
4736, 46syl5 34 . . . . . 6 ((𝐴𝐵𝑓:𝐴1-1𝐵) → (𝑤 ∈ (𝐵 ∖ ran 𝑓) → {𝐴} ≼ (𝐵 ∖ ran 𝑓)))
4847exlimdv 1901 . . . . 5 ((𝐴𝐵𝑓:𝐴1-1𝐵) → (∃𝑤 𝑤 ∈ (𝐵 ∖ ran 𝑓) → {𝐴} ≼ (𝐵 ∖ ran 𝑓)))
4935, 48mpd 15 . . . 4 ((𝐴𝐵𝑓:𝐴1-1𝐵) → {𝐴} ≼ (𝐵 ∖ ran 𝑓))
50 disjdif 4073 . . . . 5 (ran 𝑓 ∩ (𝐵 ∖ ran 𝑓)) = ∅
5150a1i 11 . . . 4 ((𝐴𝐵𝑓:𝐴1-1𝐵) → (ran 𝑓 ∩ (𝐵 ∖ ran 𝑓)) = ∅)
52 undom 8089 . . . 4 (((𝐴 ≼ ran 𝑓 ∧ {𝐴} ≼ (𝐵 ∖ ran 𝑓)) ∧ (ran 𝑓 ∩ (𝐵 ∖ ran 𝑓)) = ∅) → (𝐴 ∪ {𝐴}) ≼ (ran 𝑓 ∪ (𝐵 ∖ ran 𝑓)))
5315, 49, 51, 52syl21anc 1365 . . 3 ((𝐴𝐵𝑓:𝐴1-1𝐵) → (𝐴 ∪ {𝐴}) ≼ (ran 𝑓 ∪ (𝐵 ∖ ran 𝑓)))
54 df-suc 5767 . . . 4 suc 𝐴 = (𝐴 ∪ {𝐴})
5554a1i 11 . . 3 ((𝐴𝐵𝑓:𝐴1-1𝐵) → suc 𝐴 = (𝐴 ∪ {𝐴}))
56 undif2 4077 . . . 4 (ran 𝑓 ∪ (𝐵 ∖ ran 𝑓)) = (ran 𝑓𝐵)
5723adantl 481 . . . . 5 ((𝐴𝐵𝑓:𝐴1-1𝐵) → ran 𝑓𝐵)
58 ssequn1 3816 . . . . 5 (ran 𝑓𝐵 ↔ (ran 𝑓𝐵) = 𝐵)
5957, 58sylib 208 . . . 4 ((𝐴𝐵𝑓:𝐴1-1𝐵) → (ran 𝑓𝐵) = 𝐵)
6056, 59syl5req 2698 . . 3 ((𝐴𝐵𝑓:𝐴1-1𝐵) → 𝐵 = (ran 𝑓 ∪ (𝐵 ∖ ran 𝑓)))
6153, 55, 603brtr4d 4717 . 2 ((𝐴𝐵𝑓:𝐴1-1𝐵) → suc 𝐴𝐵)
623, 61exlimddv 1903 1 (𝐴𝐵 → suc 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wex 1744  wcel 2030  Vcvv 3231  cdif 3604  cun 3605  cin 3606  wss 3607  c0 3948  {csn 4210   class class class wbr 4685  ran crn 5144  suc csuc 5763   Fn wfn 5921  wf 5922  1-1wf1 5923  1-1-ontowf1o 5925  cen 7994  cdom 7995  csdm 7996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-suc 5767  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-1o 7605  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000
This theorem is referenced by:  sucdom  8198  card2inf  8501
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