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Theorem sucdom2 9208
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 8978 . . 3 (𝐴 β‰Ί 𝐡 β†’ 𝐴 β‰Ό 𝐡)
2 brdomi 8956 . . 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 3476 . . . . 5 𝑓 ∈ V
54rnex 7905 . . . . 5 ran 𝑓 ∈ V
6 f1f1orn 6843 . . . . . . 7 (𝑓:𝐴–1-1→𝐡 β†’ 𝑓:𝐴–1-1-ontoβ†’ran 𝑓)
76adantl 480 . . . . . 6 ((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) β†’ 𝑓:𝐴–1-1-ontoβ†’ran 𝑓)
8 f1of1 6831 . . . . . 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 8965 . . . . 5 ((𝑓 ∈ V ∧ ran 𝑓 ∈ V ∧ 𝑓:𝐴–1-1β†’ran 𝑓) β†’ 𝐴 β‰Ό ran 𝑓)
114, 5, 9, 10mp3an12i 1463 . . . 4 ((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) β†’ 𝐴 β‰Ό ran 𝑓)
12 sdomnen 8979 . . . . . . . 8 (𝐴 β‰Ί 𝐡 β†’ Β¬ 𝐴 β‰ˆ 𝐡)
1312adantr 479 . . . . . . 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 765 . . . . . . . . . . 11 (((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) ∧ 𝐡 βŠ† ran 𝑓) β†’ 𝑓:𝐴–1-1→𝐡)
16 f1f 6786 . . . . . . . . . . . . . 14 (𝑓:𝐴–1-1→𝐡 β†’ 𝑓:𝐴⟢𝐡)
1716frnd 6724 . . . . . . . . . . . . 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 483 . . . . . . . . . . . 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 6841 . . . . . . . . . . 11 (𝑓:𝐴–1-1-onto→𝐡 ↔ (𝑓:𝐴–1-1→𝐡 ∧ ran 𝑓 = 𝐡))
2215, 20, 21sylanbrc 581 . . . . . . . . . 10 (((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) ∧ 𝐡 βŠ† ran 𝑓) β†’ 𝑓:𝐴–1-1-onto→𝐡)
23 f1oen3g 8964 . . . . . . . . . 10 ((𝑓 ∈ V ∧ 𝑓:𝐴–1-1-onto→𝐡) β†’ 𝐴 β‰ˆ 𝐡)
244, 22, 23sylancr 585 . . . . . . . . 9 (((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) ∧ 𝐡 βŠ† ran 𝑓) β†’ 𝐴 β‰ˆ 𝐡)
2524ex 411 . . . . . . . 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 8948 . . . . . . . . . . 11 Rel β‰Ί
3231brrelex1i 5731 . . . . . . . . . 10 (𝐴 β‰Ί 𝐡 β†’ 𝐴 ∈ V)
3332adantr 479 . . . . . . . . 9 ((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) β†’ 𝐴 ∈ V)
34 vex 3476 . . . . . . . . 9 𝑀 ∈ V
35 en2sn 9043 . . . . . . . . 9 ((𝐴 ∈ V ∧ 𝑀 ∈ V) β†’ {𝐴} β‰ˆ {𝑀})
3633, 34, 35sylancl 584 . . . . . . . 8 ((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) β†’ {𝐴} β‰ˆ {𝑀})
3731brrelex2i 5732 . . . . . . . . . 10 (𝐴 β‰Ί 𝐡 β†’ 𝐡 ∈ V)
3837adantr 479 . . . . . . . . 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 9046 . . . . . . . . . . 11 {𝑀} ∈ Fin
41 ssdomfi2 9202 . . . . . . . . . . 11 (({𝑀} ∈ Fin ∧ (𝐡 βˆ– ran 𝑓) ∈ V ∧ {𝑀} βŠ† (𝐡 βˆ– ran 𝑓)) β†’ {𝑀} β‰Ό (𝐡 βˆ– ran 𝑓))
4240, 41mp3an1 1446 . . . . . . . . . 10 (((𝐡 βˆ– ran 𝑓) ∈ V ∧ {𝑀} βŠ† (𝐡 βˆ– ran 𝑓)) β†’ {𝑀} β‰Ό (𝐡 βˆ– ran 𝑓))
4342ex 411 . . . . . . . . 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 8977 . . . . . . . . 9 ({𝐴} β‰ˆ {𝑀} β†’ {𝐴} β‰Ό {𝑀})
46 domtrfi 9198 . . . . . . . . . 10 (({𝑀} ∈ Fin ∧ {𝐴} β‰Ό {𝑀} ∧ {𝑀} β‰Ό (𝐡 βˆ– ran 𝑓)) β†’ {𝐴} β‰Ό (𝐡 βˆ– ran 𝑓))
4740, 46mp3an1 1446 . . . . . . . . 9 (({𝐴} β‰Ό {𝑀} ∧ {𝑀} β‰Ό (𝐡 βˆ– ran 𝑓)) β†’ {𝐴} β‰Ό (𝐡 βˆ– ran 𝑓))
4845, 47sylan 578 . . . . . . . 8 (({𝐴} β‰ˆ {𝑀} ∧ {𝑀} β‰Ό (𝐡 βˆ– ran 𝑓)) β†’ {𝐴} β‰Ό (𝐡 βˆ– ran 𝑓))
4936, 44, 48syl6an 680 . . . . . . 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 1934 . . . . 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 9061 . . . 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 834 . . 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 6369 . . . 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 480 . . . . 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 2783 . . 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 1936 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 394   = wceq 1539  βˆƒwex 1779   ∈ wcel 2104  Vcvv 3472   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 6365  β€“1-1β†’wf1 6539  β€“1-1-ontoβ†’wf1o 6541   β‰ˆ cen 8938   β‰Ό cdom 8939   β‰Ί csdm 8940  Fincfn 8941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  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 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-om 7858  df-1o 8468  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945
This theorem is referenced by:  sucdom  9237  sucdomOLD  9238  card2inf  9552
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