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Theorem sucdom2 9269
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.) Avoid ax-pow 5383. (Revised by BTernaryTau, 4-Dec-2024.)
Assertion
Ref Expression
sucdom2 (𝐴𝐵 → suc 𝐴𝐵)

Proof of Theorem sucdom2
Dummy variables 𝑓 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sdomdom 9040 . . 3 (𝐴𝐵𝐴𝐵)
2 brdomi 9018 . . 3 (𝐴𝐵 → ∃𝑓 𝑓:𝐴1-1𝐵)
31, 2syl 17 . 2 (𝐴𝐵 → ∃𝑓 𝑓:𝐴1-1𝐵)
4 vex 3492 . . . . 5 𝑓 ∈ V
54rnex 7950 . . . . 5 ran 𝑓 ∈ V
6 f1f1orn 6873 . . . . . . 7 (𝑓:𝐴1-1𝐵𝑓:𝐴1-1-onto→ran 𝑓)
76adantl 481 . . . . . 6 ((𝐴𝐵𝑓:𝐴1-1𝐵) → 𝑓:𝐴1-1-onto→ran 𝑓)
8 f1of1 6861 . . . . . 6 (𝑓:𝐴1-1-onto→ran 𝑓𝑓:𝐴1-1→ran 𝑓)
97, 8syl 17 . . . . 5 ((𝐴𝐵𝑓:𝐴1-1𝐵) → 𝑓:𝐴1-1→ran 𝑓)
10 f1dom3g 9027 . . . . 5 ((𝑓 ∈ V ∧ ran 𝑓 ∈ V ∧ 𝑓:𝐴1-1→ran 𝑓) → 𝐴 ≼ ran 𝑓)
114, 5, 9, 10mp3an12i 1465 . . . 4 ((𝐴𝐵𝑓:𝐴1-1𝐵) → 𝐴 ≼ ran 𝑓)
12 sdomnen 9041 . . . . . . . 8 (𝐴𝐵 → ¬ 𝐴𝐵)
1312adantr 480 . . . . . . 7 ((𝐴𝐵𝑓:𝐴1-1𝐵) → ¬ 𝐴𝐵)
14 ssdif0 4389 . . . . . . . 8 (𝐵 ⊆ ran 𝑓 ↔ (𝐵 ∖ ran 𝑓) = ∅)
15 simplr 768 . . . . . . . . . . 11 (((𝐴𝐵𝑓:𝐴1-1𝐵) ∧ 𝐵 ⊆ ran 𝑓) → 𝑓:𝐴1-1𝐵)
16 f1f 6817 . . . . . . . . . . . . . 14 (𝑓:𝐴1-1𝐵𝑓:𝐴𝐵)
1716frnd 6755 . . . . . . . . . . . . 13 (𝑓:𝐴1-1𝐵 → ran 𝑓𝐵)
1815, 17syl 17 . . . . . . . . . . . 12 (((𝐴𝐵𝑓:𝐴1-1𝐵) ∧ 𝐵 ⊆ ran 𝑓) → ran 𝑓𝐵)
19 simpr 484 . . . . . . . . . . . 12 (((𝐴𝐵𝑓:𝐴1-1𝐵) ∧ 𝐵 ⊆ ran 𝑓) → 𝐵 ⊆ ran 𝑓)
2018, 19eqssd 4026 . . . . . . . . . . 11 (((𝐴𝐵𝑓:𝐴1-1𝐵) ∧ 𝐵 ⊆ ran 𝑓) → ran 𝑓 = 𝐵)
21 dff1o5 6871 . . . . . . . . . . 11 (𝑓:𝐴1-1-onto𝐵 ↔ (𝑓:𝐴1-1𝐵 ∧ ran 𝑓 = 𝐵))
2215, 20, 21sylanbrc 582 . . . . . . . . . 10 (((𝐴𝐵𝑓:𝐴1-1𝐵) ∧ 𝐵 ⊆ ran 𝑓) → 𝑓:𝐴1-1-onto𝐵)
23 f1oen3g 9026 . . . . . . . . . 10 ((𝑓 ∈ V ∧ 𝑓:𝐴1-1-onto𝐵) → 𝐴𝐵)
244, 22, 23sylancr 586 . . . . . . . . 9 (((𝐴𝐵𝑓:𝐴1-1𝐵) ∧ 𝐵 ⊆ ran 𝑓) → 𝐴𝐵)
2524ex 412 . . . . . . . 8 ((𝐴𝐵𝑓:𝐴1-1𝐵) → (𝐵 ⊆ ran 𝑓𝐴𝐵))
2614, 25biimtrrid 243 . . . . . . 7 ((𝐴𝐵𝑓:𝐴1-1𝐵) → ((𝐵 ∖ ran 𝑓) = ∅ → 𝐴𝐵))
2713, 26mtod 198 . . . . . 6 ((𝐴𝐵𝑓:𝐴1-1𝐵) → ¬ (𝐵 ∖ ran 𝑓) = ∅)
28 neq0 4375 . . . . . 6 (¬ (𝐵 ∖ ran 𝑓) = ∅ ↔ ∃𝑤 𝑤 ∈ (𝐵 ∖ ran 𝑓))
2927, 28sylib 218 . . . . 5 ((𝐴𝐵𝑓:𝐴1-1𝐵) → ∃𝑤 𝑤 ∈ (𝐵 ∖ ran 𝑓))
30 snssi 4833 . . . . . . 7 (𝑤 ∈ (𝐵 ∖ ran 𝑓) → {𝑤} ⊆ (𝐵 ∖ ran 𝑓))
31 relsdom 9010 . . . . . . . . . . 11 Rel ≺
3231brrelex1i 5756 . . . . . . . . . 10 (𝐴𝐵𝐴 ∈ V)
3332adantr 480 . . . . . . . . 9 ((𝐴𝐵𝑓:𝐴1-1𝐵) → 𝐴 ∈ V)
34 vex 3492 . . . . . . . . 9 𝑤 ∈ V
35 en2sn 9106 . . . . . . . . 9 ((𝐴 ∈ V ∧ 𝑤 ∈ V) → {𝐴} ≈ {𝑤})
3633, 34, 35sylancl 585 . . . . . . . 8 ((𝐴𝐵𝑓:𝐴1-1𝐵) → {𝐴} ≈ {𝑤})
3731brrelex2i 5757 . . . . . . . . . 10 (𝐴𝐵𝐵 ∈ V)
3837adantr 480 . . . . . . . . 9 ((𝐴𝐵𝑓:𝐴1-1𝐵) → 𝐵 ∈ V)
39 difexg 5347 . . . . . . . . 9 (𝐵 ∈ V → (𝐵 ∖ ran 𝑓) ∈ V)
40 snfi 9109 . . . . . . . . . . 11 {𝑤} ∈ Fin
41 ssdomfi2 9263 . . . . . . . . . . 11 (({𝑤} ∈ Fin ∧ (𝐵 ∖ ran 𝑓) ∈ V ∧ {𝑤} ⊆ (𝐵 ∖ ran 𝑓)) → {𝑤} ≼ (𝐵 ∖ ran 𝑓))
4240, 41mp3an1 1448 . . . . . . . . . 10 (((𝐵 ∖ ran 𝑓) ∈ V ∧ {𝑤} ⊆ (𝐵 ∖ ran 𝑓)) → {𝑤} ≼ (𝐵 ∖ ran 𝑓))
4342ex 412 . . . . . . . . 9 ((𝐵 ∖ ran 𝑓) ∈ V → ({𝑤} ⊆ (𝐵 ∖ ran 𝑓) → {𝑤} ≼ (𝐵 ∖ ran 𝑓)))
4438, 39, 433syl 18 . . . . . . . 8 ((𝐴𝐵𝑓:𝐴1-1𝐵) → ({𝑤} ⊆ (𝐵 ∖ ran 𝑓) → {𝑤} ≼ (𝐵 ∖ ran 𝑓)))
45 endom 9039 . . . . . . . . 9 ({𝐴} ≈ {𝑤} → {𝐴} ≼ {𝑤})
46 domtrfi 9259 . . . . . . . . . 10 (({𝑤} ∈ Fin ∧ {𝐴} ≼ {𝑤} ∧ {𝑤} ≼ (𝐵 ∖ ran 𝑓)) → {𝐴} ≼ (𝐵 ∖ ran 𝑓))
4740, 46mp3an1 1448 . . . . . . . . 9 (({𝐴} ≼ {𝑤} ∧ {𝑤} ≼ (𝐵 ∖ ran 𝑓)) → {𝐴} ≼ (𝐵 ∖ ran 𝑓))
4845, 47sylan 579 . . . . . . . 8 (({𝐴} ≈ {𝑤} ∧ {𝑤} ≼ (𝐵 ∖ ran 𝑓)) → {𝐴} ≼ (𝐵 ∖ ran 𝑓))
4936, 44, 48syl6an 683 . . . . . . 7 ((𝐴𝐵𝑓:𝐴1-1𝐵) → ({𝑤} ⊆ (𝐵 ∖ ran 𝑓) → {𝐴} ≼ (𝐵 ∖ ran 𝑓)))
5030, 49syl5 34 . . . . . 6 ((𝐴𝐵𝑓:𝐴1-1𝐵) → (𝑤 ∈ (𝐵 ∖ ran 𝑓) → {𝐴} ≼ (𝐵 ∖ ran 𝑓)))
5150exlimdv 1932 . . . . 5 ((𝐴𝐵𝑓:𝐴1-1𝐵) → (∃𝑤 𝑤 ∈ (𝐵 ∖ ran 𝑓) → {𝐴} ≼ (𝐵 ∖ ran 𝑓)))
5229, 51mpd 15 . . . 4 ((𝐴𝐵𝑓:𝐴1-1𝐵) → {𝐴} ≼ (𝐵 ∖ ran 𝑓))
53 disjdif 4495 . . . . 5 (ran 𝑓 ∩ (𝐵 ∖ ran 𝑓)) = ∅
5453a1i 11 . . . 4 ((𝐴𝐵𝑓:𝐴1-1𝐵) → (ran 𝑓 ∩ (𝐵 ∖ ran 𝑓)) = ∅)
55 undom 9125 . . . 4 (((𝐴 ≼ ran 𝑓 ∧ {𝐴} ≼ (𝐵 ∖ ran 𝑓)) ∧ (ran 𝑓 ∩ (𝐵 ∖ ran 𝑓)) = ∅) → (𝐴 ∪ {𝐴}) ≼ (ran 𝑓 ∪ (𝐵 ∖ ran 𝑓)))
5611, 52, 54, 55syl21anc 837 . . 3 ((𝐴𝐵𝑓:𝐴1-1𝐵) → (𝐴 ∪ {𝐴}) ≼ (ran 𝑓 ∪ (𝐵 ∖ ran 𝑓)))
57 df-suc 6401 . . . 4 suc 𝐴 = (𝐴 ∪ {𝐴})
5857a1i 11 . . 3 ((𝐴𝐵𝑓:𝐴1-1𝐵) → suc 𝐴 = (𝐴 ∪ {𝐴}))
59 undif2 4500 . . . 4 (ran 𝑓 ∪ (𝐵 ∖ ran 𝑓)) = (ran 𝑓𝐵)
6017adantl 481 . . . . 5 ((𝐴𝐵𝑓:𝐴1-1𝐵) → ran 𝑓𝐵)
61 ssequn1 4209 . . . . 5 (ran 𝑓𝐵 ↔ (ran 𝑓𝐵) = 𝐵)
6260, 61sylib 218 . . . 4 ((𝐴𝐵𝑓:𝐴1-1𝐵) → (ran 𝑓𝐵) = 𝐵)
6359, 62eqtr2id 2793 . . 3 ((𝐴𝐵𝑓:𝐴1-1𝐵) → 𝐵 = (ran 𝑓 ∪ (𝐵 ∖ ran 𝑓)))
6456, 58, 633brtr4d 5198 . 2 ((𝐴𝐵𝑓:𝐴1-1𝐵) → suc 𝐴𝐵)
653, 64exlimddv 1934 1 (𝐴𝐵 → suc 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wex 1777  wcel 2108  Vcvv 3488  cdif 3973  cun 3974  cin 3975  wss 3976  c0 4352  {csn 4648   class class class wbr 5166  ran crn 5701  suc csuc 6397  1-1wf1 6570  1-1-ontowf1o 6572  cen 9000  cdom 9001  csdm 9002  Fincfn 9003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-om 7904  df-1o 8522  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007
This theorem is referenced by:  sucdom  9298  sucdomOLD  9299  card2inf  9624
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