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Theorem sucdom2 8363
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 8188 . . 3 (𝐴𝐵𝐴𝐵)
2 brdomi 8171 . . 3 (𝐴𝐵 → ∃𝑓 𝑓:𝐴1-1𝐵)
31, 2syl 17 . 2 (𝐴𝐵 → ∃𝑓 𝑓:𝐴1-1𝐵)
4 relsdom 8167 . . . . . . 7 Rel ≺
54brrelex1i 5328 . . . . . 6 (𝐴𝐵𝐴 ∈ V)
65adantr 472 . . . . 5 ((𝐴𝐵𝑓:𝐴1-1𝐵) → 𝐴 ∈ V)
7 vex 3353 . . . . . . 7 𝑓 ∈ V
87rnex 7298 . . . . . 6 ran 𝑓 ∈ V
98a1i 11 . . . . 5 ((𝐴𝐵𝑓:𝐴1-1𝐵) → ran 𝑓 ∈ V)
10 f1f1orn 6331 . . . . . . 7 (𝑓:𝐴1-1𝐵𝑓:𝐴1-1-onto→ran 𝑓)
1110adantl 473 . . . . . 6 ((𝐴𝐵𝑓:𝐴1-1𝐵) → 𝑓:𝐴1-1-onto→ran 𝑓)
12 f1of1 6319 . . . . . 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 8178 . . . . 5 ((𝐴 ∈ V ∧ ran 𝑓 ∈ V ∧ 𝑓:𝐴1-1→ran 𝑓) → 𝐴 ≼ ran 𝑓)
156, 9, 13, 14syl3anc 1490 . . . 4 ((𝐴𝐵𝑓:𝐴1-1𝐵) → 𝐴 ≼ ran 𝑓)
16 sdomnen 8189 . . . . . . . 8 (𝐴𝐵 → ¬ 𝐴𝐵)
1716adantr 472 . . . . . . 7 ((𝐴𝐵𝑓:𝐴1-1𝐵) → ¬ 𝐴𝐵)
18 ssdif0 4106 . . . . . . . 8 (𝐵 ⊆ ran 𝑓 ↔ (𝐵 ∖ ran 𝑓) = ∅)
19 simplr 785 . . . . . . . . . . 11 (((𝐴𝐵𝑓:𝐴1-1𝐵) ∧ 𝐵 ⊆ ran 𝑓) → 𝑓:𝐴1-1𝐵)
20 f1f 6283 . . . . . . . . . . . . . . 15 (𝑓:𝐴1-1𝐵𝑓:𝐴𝐵)
21 df-f 6072 . . . . . . . . . . . . . . 15 (𝑓:𝐴𝐵 ↔ (𝑓 Fn 𝐴 ∧ ran 𝑓𝐵))
2220, 21sylib 209 . . . . . . . . . . . . . 14 (𝑓:𝐴1-1𝐵 → (𝑓 Fn 𝐴 ∧ ran 𝑓𝐵))
2322simprd 489 . . . . . . . . . . . . 13 (𝑓:𝐴1-1𝐵 → ran 𝑓𝐵)
2419, 23syl 17 . . . . . . . . . . . 12 (((𝐴𝐵𝑓:𝐴1-1𝐵) ∧ 𝐵 ⊆ ran 𝑓) → ran 𝑓𝐵)
25 simpr 477 . . . . . . . . . . . 12 (((𝐴𝐵𝑓:𝐴1-1𝐵) ∧ 𝐵 ⊆ ran 𝑓) → 𝐵 ⊆ ran 𝑓)
2624, 25eqssd 3778 . . . . . . . . . . 11 (((𝐴𝐵𝑓:𝐴1-1𝐵) ∧ 𝐵 ⊆ ran 𝑓) → ran 𝑓 = 𝐵)
27 dff1o5 6329 . . . . . . . . . . 11 (𝑓:𝐴1-1-onto𝐵 ↔ (𝑓:𝐴1-1𝐵 ∧ ran 𝑓 = 𝐵))
2819, 26, 27sylanbrc 578 . . . . . . . . . 10 (((𝐴𝐵𝑓:𝐴1-1𝐵) ∧ 𝐵 ⊆ ran 𝑓) → 𝑓:𝐴1-1-onto𝐵)
29 f1oen3g 8176 . . . . . . . . . 10 ((𝑓 ∈ V ∧ 𝑓:𝐴1-1-onto𝐵) → 𝐴𝐵)
307, 28, 29sylancr 581 . . . . . . . . 9 (((𝐴𝐵𝑓:𝐴1-1𝐵) ∧ 𝐵 ⊆ ran 𝑓) → 𝐴𝐵)
3130ex 401 . . . . . . . 8 ((𝐴𝐵𝑓:𝐴1-1𝐵) → (𝐵 ⊆ ran 𝑓𝐴𝐵))
3218, 31syl5bir 234 . . . . . . 7 ((𝐴𝐵𝑓:𝐴1-1𝐵) → ((𝐵 ∖ ran 𝑓) = ∅ → 𝐴𝐵))
3317, 32mtod 189 . . . . . 6 ((𝐴𝐵𝑓:𝐴1-1𝐵) → ¬ (𝐵 ∖ ran 𝑓) = ∅)
34 neq0 4094 . . . . . 6 (¬ (𝐵 ∖ ran 𝑓) = ∅ ↔ ∃𝑤 𝑤 ∈ (𝐵 ∖ ran 𝑓))
3533, 34sylib 209 . . . . 5 ((𝐴𝐵𝑓:𝐴1-1𝐵) → ∃𝑤 𝑤 ∈ (𝐵 ∖ ran 𝑓))
36 snssi 4493 . . . . . . 7 (𝑤 ∈ (𝐵 ∖ ran 𝑓) → {𝑤} ⊆ (𝐵 ∖ ran 𝑓))
37 vex 3353 . . . . . . . . 9 𝑤 ∈ V
38 en2sn 8244 . . . . . . . . 9 ((𝐴 ∈ V ∧ 𝑤 ∈ V) → {𝐴} ≈ {𝑤})
396, 37, 38sylancl 580 . . . . . . . 8 ((𝐴𝐵𝑓:𝐴1-1𝐵) → {𝐴} ≈ {𝑤})
404brrelex2i 5329 . . . . . . . . . 10 (𝐴𝐵𝐵 ∈ V)
4140adantr 472 . . . . . . . . 9 ((𝐴𝐵𝑓:𝐴1-1𝐵) → 𝐵 ∈ V)
42 difexg 4969 . . . . . . . . 9 (𝐵 ∈ V → (𝐵 ∖ ran 𝑓) ∈ V)
43 ssdomg 8206 . . . . . . . . 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 8218 . . . . . . . 8 (({𝐴} ≈ {𝑤} ∧ {𝑤} ≼ (𝐵 ∖ ran 𝑓)) → {𝐴} ≼ (𝐵 ∖ ran 𝑓))
4639, 44, 45syl6an 674 . . . . . . 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 2028 . . . . 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 4200 . . . . 5 (ran 𝑓 ∩ (𝐵 ∖ ran 𝑓)) = ∅
5150a1i 11 . . . 4 ((𝐴𝐵𝑓:𝐴1-1𝐵) → (ran 𝑓 ∩ (𝐵 ∖ ran 𝑓)) = ∅)
52 undom 8255 . . . 4 (((𝐴 ≼ ran 𝑓 ∧ {𝐴} ≼ (𝐵 ∖ ran 𝑓)) ∧ (ran 𝑓 ∩ (𝐵 ∖ ran 𝑓)) = ∅) → (𝐴 ∪ {𝐴}) ≼ (ran 𝑓 ∪ (𝐵 ∖ ran 𝑓)))
5315, 49, 51, 52syl21anc 866 . . 3 ((𝐴𝐵𝑓:𝐴1-1𝐵) → (𝐴 ∪ {𝐴}) ≼ (ran 𝑓 ∪ (𝐵 ∖ ran 𝑓)))
54 df-suc 5914 . . . 4 suc 𝐴 = (𝐴 ∪ {𝐴})
5554a1i 11 . . 3 ((𝐴𝐵𝑓:𝐴1-1𝐵) → suc 𝐴 = (𝐴 ∪ {𝐴}))
56 undif2 4204 . . . 4 (ran 𝑓 ∪ (𝐵 ∖ ran 𝑓)) = (ran 𝑓𝐵)
5723adantl 473 . . . . 5 ((𝐴𝐵𝑓:𝐴1-1𝐵) → ran 𝑓𝐵)
58 ssequn1 3945 . . . . 5 (ran 𝑓𝐵 ↔ (ran 𝑓𝐵) = 𝐵)
5957, 58sylib 209 . . . 4 ((𝐴𝐵𝑓:𝐴1-1𝐵) → (ran 𝑓𝐵) = 𝐵)
6056, 59syl5req 2812 . . 3 ((𝐴𝐵𝑓:𝐴1-1𝐵) → 𝐵 = (ran 𝑓 ∪ (𝐵 ∖ ran 𝑓)))
6153, 55, 603brtr4d 4841 . 2 ((𝐴𝐵𝑓:𝐴1-1𝐵) → suc 𝐴𝐵)
623, 61exlimddv 2030 1 (𝐴𝐵 → suc 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1652  wex 1874  wcel 2155  Vcvv 3350  cdif 3729  cun 3730  cin 3731  wss 3732  c0 4079  {csn 4334   class class class wbr 4809  ran crn 5278  suc csuc 5910   Fn wfn 6063  wf 6064  1-1wf1 6065  1-1-ontowf1o 6067  cen 8157  cdom 8158  csdm 8159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-suc 5914  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-1o 7764  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163
This theorem is referenced by:  sucdom  8364  card2inf  8667
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