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Theorem sucdom2 9202
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 5362. (Revised by BTernaryTau, 4-Dec-2024.)
Assertion
Ref Expression
sucdom2 (𝐴 β‰Ί 𝐡 β†’ suc 𝐴 β‰Ό 𝐡)
Proof of Theorem sucdom2
Dummy variables 𝑓 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sdomdom 8972 . . 3 (𝐴 β‰Ί 𝐡 β†’ 𝐴 β‰Ό 𝐡)
2 brdomi 8950 . . 3 (𝐴 β‰Ό 𝐡 β†’ βˆƒπ‘“ 𝑓:𝐴–1-1→𝐡)
31, 2syl 17 . 2 (𝐴 β‰Ί 𝐡 β†’ βˆƒπ‘“ 𝑓:𝐴–1-1→𝐡)
4 vex 3478 . . . . 5 𝑓 ∈ V
54rnex 7899 . . . . 5 ran 𝑓 ∈ V
6 f1f1orn 6841 . . . . . . 7 (𝑓:𝐴–1-1→𝐡 β†’ 𝑓:𝐴–1-1-ontoβ†’ran 𝑓)
76adantl 482 . . . . . 6 ((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) β†’ 𝑓:𝐴–1-1-ontoβ†’ran 𝑓)
8 f1of1 6829 . . . . . 6 (𝑓:𝐴–1-1-ontoβ†’ran 𝑓 β†’ 𝑓:𝐴–1-1β†’ran 𝑓)
97, 8syl 17 . . . . 5 ((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) β†’ 𝑓:𝐴–1-1β†’ran 𝑓)
10 f1dom3g 8959 . . . . 5 ((𝑓 ∈ V ∧ ran 𝑓 ∈ V ∧ 𝑓:𝐴–1-1β†’ran 𝑓) β†’ 𝐴 β‰Ό ran 𝑓)
114, 5, 9, 10mp3an12i 1465 . . . 4 ((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) β†’ 𝐴 β‰Ό ran 𝑓)
12 sdomnen 8973 . . . . . . . 8 (𝐴 β‰Ί 𝐡 β†’ Β¬ 𝐴 β‰ˆ 𝐡)
1312adantr 481 . . . . . . 7 ((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) β†’ Β¬ 𝐴 β‰ˆ 𝐡)
14 ssdif0 4362 . . . . . . . 8 (𝐡 βŠ† ran 𝑓 ↔ (𝐡 βˆ– ran 𝑓) = βˆ…)
15 simplr 767 . . . . . . . . . . 11 (((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) ∧ 𝐡 βŠ† ran 𝑓) β†’ 𝑓:𝐴–1-1→𝐡)
16 f1f 6784 . . . . . . . . . . . . . 14 (𝑓:𝐴–1-1→𝐡 β†’ 𝑓:𝐴⟢𝐡)
1716frnd 6722 . . . . . . . . . . . . 13 (𝑓:𝐴–1-1→𝐡 β†’ ran 𝑓 βŠ† 𝐡)
1815, 17syl 17 . . . . . . . . . . . 12 (((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) ∧ 𝐡 βŠ† ran 𝑓) β†’ ran 𝑓 βŠ† 𝐡)
19 simpr 485 . . . . . . . . . . . 12 (((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) ∧ 𝐡 βŠ† ran 𝑓) β†’ 𝐡 βŠ† ran 𝑓)
2018, 19eqssd 3998 . . . . . . . . . . 11 (((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) ∧ 𝐡 βŠ† ran 𝑓) β†’ ran 𝑓 = 𝐡)
21 dff1o5 6839 . . . . . . . . . . 11 (𝑓:𝐴–1-1-onto→𝐡 ↔ (𝑓:𝐴–1-1→𝐡 ∧ ran 𝑓 = 𝐡))
2215, 20, 21sylanbrc 583 . . . . . . . . . 10 (((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) ∧ 𝐡 βŠ† ran 𝑓) β†’ 𝑓:𝐴–1-1-onto→𝐡)
23 f1oen3g 8958 . . . . . . . . . 10 ((𝑓 ∈ V ∧ 𝑓:𝐴–1-1-onto→𝐡) β†’ 𝐴 β‰ˆ 𝐡)
244, 22, 23sylancr 587 . . . . . . . . 9 (((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) ∧ 𝐡 βŠ† ran 𝑓) β†’ 𝐴 β‰ˆ 𝐡)
2524ex 413 . . . . . . . 8 ((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) β†’ (𝐡 βŠ† ran 𝑓 β†’ 𝐴 β‰ˆ 𝐡))
2614, 25biimtrrid 242 . . . . . . 7 ((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) β†’ ((𝐡 βˆ– ran 𝑓) = βˆ… β†’ 𝐴 β‰ˆ 𝐡))
2713, 26mtod 197 . . . . . 6 ((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) β†’ Β¬ (𝐡 βˆ– ran 𝑓) = βˆ…)
28 neq0 4344 . . . . . 6 (Β¬ (𝐡 βˆ– ran 𝑓) = βˆ… ↔ βˆƒπ‘€ 𝑀 ∈ (𝐡 βˆ– ran 𝑓))
2927, 28sylib 217 . . . . 5 ((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) β†’ βˆƒπ‘€ 𝑀 ∈ (𝐡 βˆ– ran 𝑓))
30 snssi 4810 . . . . . . 7 (𝑀 ∈ (𝐡 βˆ– ran 𝑓) β†’ {𝑀} βŠ† (𝐡 βˆ– ran 𝑓))
31 relsdom 8942 . . . . . . . . . . 11 Rel β‰Ί
3231brrelex1i 5730 . . . . . . . . . 10 (𝐴 β‰Ί 𝐡 β†’ 𝐴 ∈ V)
3332adantr 481 . . . . . . . . 9 ((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) β†’ 𝐴 ∈ V)
34 vex 3478 . . . . . . . . 9 𝑀 ∈ V
35 en2sn 9037 . . . . . . . . 9 ((𝐴 ∈ V ∧ 𝑀 ∈ V) β†’ {𝐴} β‰ˆ {𝑀})
3633, 34, 35sylancl 586 . . . . . . . 8 ((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) β†’ {𝐴} β‰ˆ {𝑀})
3731brrelex2i 5731 . . . . . . . . . 10 (𝐴 β‰Ί 𝐡 β†’ 𝐡 ∈ V)
3837adantr 481 . . . . . . . . 9 ((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) β†’ 𝐡 ∈ V)
39 difexg 5326 . . . . . . . . 9 (𝐡 ∈ V β†’ (𝐡 βˆ– ran 𝑓) ∈ V)
40 snfi 9040 . . . . . . . . . . 11 {𝑀} ∈ Fin
41 ssdomfi2 9196 . . . . . . . . . . 11 (({𝑀} ∈ Fin ∧ (𝐡 βˆ– ran 𝑓) ∈ V ∧ {𝑀} βŠ† (𝐡 βˆ– ran 𝑓)) β†’ {𝑀} β‰Ό (𝐡 βˆ– ran 𝑓))
4240, 41mp3an1 1448 . . . . . . . . . 10 (((𝐡 βˆ– ran 𝑓) ∈ V ∧ {𝑀} βŠ† (𝐡 βˆ– ran 𝑓)) β†’ {𝑀} β‰Ό (𝐡 βˆ– ran 𝑓))
4342ex 413 . . . . . . . . 9 ((𝐡 βˆ– ran 𝑓) ∈ V β†’ ({𝑀} βŠ† (𝐡 βˆ– ran 𝑓) β†’ {𝑀} β‰Ό (𝐡 βˆ– ran 𝑓)))
4438, 39, 433syl 18 . . . . . . . 8 ((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) β†’ ({𝑀} βŠ† (𝐡 βˆ– ran 𝑓) β†’ {𝑀} β‰Ό (𝐡 βˆ– ran 𝑓)))
45 endom 8971 . . . . . . . . 9 ({𝐴} β‰ˆ {𝑀} β†’ {𝐴} β‰Ό {𝑀})
46 domtrfi 9192 . . . . . . . . . 10 (({𝑀} ∈ Fin ∧ {𝐴} β‰Ό {𝑀} ∧ {𝑀} β‰Ό (𝐡 βˆ– ran 𝑓)) β†’ {𝐴} β‰Ό (𝐡 βˆ– ran 𝑓))
4740, 46mp3an1 1448 . . . . . . . . 9 (({𝐴} β‰Ό {𝑀} ∧ {𝑀} β‰Ό (𝐡 βˆ– ran 𝑓)) β†’ {𝐴} β‰Ό (𝐡 βˆ– ran 𝑓))
4845, 47sylan 580 . . . . . . . 8 (({𝐴} β‰ˆ {𝑀} ∧ {𝑀} β‰Ό (𝐡 βˆ– ran 𝑓)) β†’ {𝐴} β‰Ό (𝐡 βˆ– ran 𝑓))
4936, 44, 48syl6an 682 . . . . . . 7 ((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) β†’ ({𝑀} βŠ† (𝐡 βˆ– ran 𝑓) β†’ {𝐴} β‰Ό (𝐡 βˆ– ran 𝑓)))
5030, 49syl5 34 . . . . . 6 ((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) β†’ (𝑀 ∈ (𝐡 βˆ– ran 𝑓) β†’ {𝐴} β‰Ό (𝐡 βˆ– ran 𝑓)))
5150exlimdv 1936 . . . . 5 ((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) β†’ (βˆƒπ‘€ 𝑀 ∈ (𝐡 βˆ– ran 𝑓) β†’ {𝐴} β‰Ό (𝐡 βˆ– ran 𝑓)))
5229, 51mpd 15 . . . 4 ((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) β†’ {𝐴} β‰Ό (𝐡 βˆ– ran 𝑓))
53 disjdif 4470 . . . . 5 (ran 𝑓 ∩ (𝐡 βˆ– ran 𝑓)) = βˆ…
5453a1i 11 . . . 4 ((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) β†’ (ran 𝑓 ∩ (𝐡 βˆ– ran 𝑓)) = βˆ…)
55 undom 9055 . . . 4 (((𝐴 β‰Ό ran 𝑓 ∧ {𝐴} β‰Ό (𝐡 βˆ– ran 𝑓)) ∧ (ran 𝑓 ∩ (𝐡 βˆ– ran 𝑓)) = βˆ…) β†’ (𝐴 βˆͺ {𝐴}) β‰Ό (ran 𝑓 βˆͺ (𝐡 βˆ– ran 𝑓)))
5611, 52, 54, 55syl21anc 836 . . 3 ((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) β†’ (𝐴 βˆͺ {𝐴}) β‰Ό (ran 𝑓 βˆͺ (𝐡 βˆ– ran 𝑓)))
57 df-suc 6367 . . . 4 suc 𝐴 = (𝐴 βˆͺ {𝐴})
5857a1i 11 . . 3 ((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) β†’ suc 𝐴 = (𝐴 βˆͺ {𝐴}))
59 undif2 4475 . . . 4 (ran 𝑓 βˆͺ (𝐡 βˆ– ran 𝑓)) = (ran 𝑓 βˆͺ 𝐡)
6017adantl 482 . . . . 5 ((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) β†’ ran 𝑓 βŠ† 𝐡)
61 ssequn1 4179 . . . . 5 (ran 𝑓 βŠ† 𝐡 ↔ (ran 𝑓 βˆͺ 𝐡) = 𝐡)
6260, 61sylib 217 . . . 4 ((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) β†’ (ran 𝑓 βˆͺ 𝐡) = 𝐡)
6359, 62eqtr2id 2785 . . 3 ((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) β†’ 𝐡 = (ran 𝑓 βˆͺ (𝐡 βˆ– ran 𝑓)))
6456, 58, 633brtr4d 5179 . 2 ((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) β†’ suc 𝐴 β‰Ό 𝐡)
653, 64exlimddv 1938 1 (𝐴 β‰Ί 𝐡 β†’ suc 𝐴 β‰Ό 𝐡)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 396   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106  Vcvv 3474   βˆ– cdif 3944   βˆͺ cun 3945   ∩ cin 3946   βŠ† wss 3947  βˆ…c0 4321  {csn 4627   class class class wbr 5147  ran crn 5676  suc csuc 6363  β€“1-1β†’wf1 6537  β€“1-1-ontoβ†’wf1o 6539   β‰ˆ cen 8932   β‰Ό cdom 8933   β‰Ί csdm 8934  Fincfn 8935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-om 7852  df-1o 8462  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939
This theorem is referenced by:  sucdom  9231  sucdomOLD  9232  card2inf  9546
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