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Theorem sucdom2 9157
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 5325. (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 8927 . . 3 (𝐴 β‰Ί 𝐡 β†’ 𝐴 β‰Ό 𝐡)
2 brdomi 8905 . . 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 3452 . . . . 5 𝑓 ∈ V
54rnex 7854 . . . . 5 ran 𝑓 ∈ V
6 f1f1orn 6800 . . . . . . 7 (𝑓:𝐴–1-1→𝐡 β†’ 𝑓:𝐴–1-1-ontoβ†’ran 𝑓)
76adantl 483 . . . . . 6 ((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) β†’ 𝑓:𝐴–1-1-ontoβ†’ran 𝑓)
8 f1of1 6788 . . . . . 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 8914 . . . . 5 ((𝑓 ∈ V ∧ ran 𝑓 ∈ V ∧ 𝑓:𝐴–1-1β†’ran 𝑓) β†’ 𝐴 β‰Ό ran 𝑓)
114, 5, 9, 10mp3an12i 1466 . . . 4 ((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) β†’ 𝐴 β‰Ό ran 𝑓)
12 sdomnen 8928 . . . . . . . 8 (𝐴 β‰Ί 𝐡 β†’ Β¬ 𝐴 β‰ˆ 𝐡)
1312adantr 482 . . . . . . 7 ((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) β†’ Β¬ 𝐴 β‰ˆ 𝐡)
14 ssdif0 4328 . . . . . . . 8 (𝐡 βŠ† ran 𝑓 ↔ (𝐡 βˆ– ran 𝑓) = βˆ…)
15 simplr 768 . . . . . . . . . . 11 (((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) ∧ 𝐡 βŠ† ran 𝑓) β†’ 𝑓:𝐴–1-1→𝐡)
16 f1f 6743 . . . . . . . . . . . . . 14 (𝑓:𝐴–1-1→𝐡 β†’ 𝑓:𝐴⟢𝐡)
1716frnd 6681 . . . . . . . . . . . . 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 486 . . . . . . . . . . . 12 (((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) ∧ 𝐡 βŠ† ran 𝑓) β†’ 𝐡 βŠ† ran 𝑓)
2018, 19eqssd 3966 . . . . . . . . . . 11 (((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) ∧ 𝐡 βŠ† ran 𝑓) β†’ ran 𝑓 = 𝐡)
21 dff1o5 6798 . . . . . . . . . . 11 (𝑓:𝐴–1-1-onto→𝐡 ↔ (𝑓:𝐴–1-1→𝐡 ∧ ran 𝑓 = 𝐡))
2215, 20, 21sylanbrc 584 . . . . . . . . . 10 (((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) ∧ 𝐡 βŠ† ran 𝑓) β†’ 𝑓:𝐴–1-1-onto→𝐡)
23 f1oen3g 8913 . . . . . . . . . 10 ((𝑓 ∈ V ∧ 𝑓:𝐴–1-1-onto→𝐡) β†’ 𝐴 β‰ˆ 𝐡)
244, 22, 23sylancr 588 . . . . . . . . 9 (((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) ∧ 𝐡 βŠ† ran 𝑓) β†’ 𝐴 β‰ˆ 𝐡)
2524ex 414 . . . . . . . 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 4310 . . . . . 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 4773 . . . . . . 7 (𝑀 ∈ (𝐡 βˆ– ran 𝑓) β†’ {𝑀} βŠ† (𝐡 βˆ– ran 𝑓))
31 relsdom 8897 . . . . . . . . . . 11 Rel β‰Ί
3231brrelex1i 5693 . . . . . . . . . 10 (𝐴 β‰Ί 𝐡 β†’ 𝐴 ∈ V)
3332adantr 482 . . . . . . . . 9 ((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) β†’ 𝐴 ∈ V)
34 vex 3452 . . . . . . . . 9 𝑀 ∈ V
35 en2sn 8992 . . . . . . . . 9 ((𝐴 ∈ V ∧ 𝑀 ∈ V) β†’ {𝐴} β‰ˆ {𝑀})
3633, 34, 35sylancl 587 . . . . . . . 8 ((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) β†’ {𝐴} β‰ˆ {𝑀})
3731brrelex2i 5694 . . . . . . . . . 10 (𝐴 β‰Ί 𝐡 β†’ 𝐡 ∈ V)
3837adantr 482 . . . . . . . . 9 ((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) β†’ 𝐡 ∈ V)
39 difexg 5289 . . . . . . . . 9 (𝐡 ∈ V β†’ (𝐡 βˆ– ran 𝑓) ∈ V)
40 snfi 8995 . . . . . . . . . . 11 {𝑀} ∈ Fin
41 ssdomfi2 9151 . . . . . . . . . . 11 (({𝑀} ∈ Fin ∧ (𝐡 βˆ– ran 𝑓) ∈ V ∧ {𝑀} βŠ† (𝐡 βˆ– ran 𝑓)) β†’ {𝑀} β‰Ό (𝐡 βˆ– ran 𝑓))
4240, 41mp3an1 1449 . . . . . . . . . 10 (((𝐡 βˆ– ran 𝑓) ∈ V ∧ {𝑀} βŠ† (𝐡 βˆ– ran 𝑓)) β†’ {𝑀} β‰Ό (𝐡 βˆ– ran 𝑓))
4342ex 414 . . . . . . . . 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 8926 . . . . . . . . 9 ({𝐴} β‰ˆ {𝑀} β†’ {𝐴} β‰Ό {𝑀})
46 domtrfi 9147 . . . . . . . . . 10 (({𝑀} ∈ Fin ∧ {𝐴} β‰Ό {𝑀} ∧ {𝑀} β‰Ό (𝐡 βˆ– ran 𝑓)) β†’ {𝐴} β‰Ό (𝐡 βˆ– ran 𝑓))
4740, 46mp3an1 1449 . . . . . . . . 9 (({𝐴} β‰Ό {𝑀} ∧ {𝑀} β‰Ό (𝐡 βˆ– ran 𝑓)) β†’ {𝐴} β‰Ό (𝐡 βˆ– ran 𝑓))
4845, 47sylan 581 . . . . . . . 8 (({𝐴} β‰ˆ {𝑀} ∧ {𝑀} β‰Ό (𝐡 βˆ– ran 𝑓)) β†’ {𝐴} β‰Ό (𝐡 βˆ– ran 𝑓))
4936, 44, 48syl6an 683 . . . . . . 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 1937 . . . . 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 4436 . . . . 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 9010 . . . 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 837 . . 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 6328 . . . 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 4441 . . . 4 (ran 𝑓 βˆͺ (𝐡 βˆ– ran 𝑓)) = (ran 𝑓 βˆͺ 𝐡)
6017adantl 483 . . . . 5 ((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) β†’ ran 𝑓 βŠ† 𝐡)
61 ssequn1 4145 . . . . 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 2790 . . 3 ((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) β†’ 𝐡 = (ran 𝑓 βˆͺ (𝐡 βˆ– ran 𝑓)))
6456, 58, 633brtr4d 5142 . 2 ((𝐴 β‰Ί 𝐡 ∧ 𝑓:𝐴–1-1→𝐡) β†’ suc 𝐴 β‰Ό 𝐡)
653, 64exlimddv 1939 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 397   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  Vcvv 3448   βˆ– cdif 3912   βˆͺ cun 3913   ∩ cin 3914   βŠ† wss 3915  βˆ…c0 4287  {csn 4591   class class class wbr 5110  ran crn 5639  suc csuc 6324  β€“1-1β†’wf1 6498  β€“1-1-ontoβ†’wf1o 6500   β‰ˆ cen 8887   β‰Ό cdom 8888   β‰Ί csdm 8889  Fincfn 8890
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-om 7808  df-1o 8417  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894
This theorem is referenced by:  sucdom  9186  sucdomOLD  9187  card2inf  9498
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