Theorem sucdom2 9125

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 5308. (Revised by BTernaryTau, 4-Dec-2024.)

Assertion

Ref	Expression
sucdom2	⊢ (𝐴 ≺ 𝐵 → suc 𝐴 ≼ 𝐵)

Proof of Theorem sucdom2

Dummy variables 𝑓 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sdomdom 8915	. . 3 ⊢ (𝐴 ≺ 𝐵 → 𝐴 ≼ 𝐵)
2		brdomi 8894	. . 3 ⊢ (𝐴 ≼ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵)
3	1, 2	syl 17	. 2 ⊢ (𝐴 ≺ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵)
4		vex 3442	. . . . 5 ⊢ 𝑓 ∈ V
5	4	rnex 7850	. . . . 5 ⊢ ran 𝑓 ∈ V
6		f1f1orn 6783	. . . . . . 7 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴–1-1-onto→ran 𝑓)
7	6	adantl 481	. . . . . 6 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → 𝑓:𝐴–1-1-onto→ran 𝑓)
8		f1of1 6771	. . . . . 6 ⊢ (𝑓:𝐴–1-1-onto→ran 𝑓 → 𝑓:𝐴–1-1→ran 𝑓)
9	7, 8	syl 17	. . . . 5 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → 𝑓:𝐴–1-1→ran 𝑓)
10		f1dom3g 8902	. . . . 5 ⊢ ((𝑓 ∈ V ∧ ran 𝑓 ∈ V ∧ 𝑓:𝐴–1-1→ran 𝑓) → 𝐴 ≼ ran 𝑓)
11	4, 5, 9, 10	mp3an12i 1467	. . . 4 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → 𝐴 ≼ ran 𝑓)
12		sdomnen 8916	. . . . . . . 8 ⊢ (𝐴 ≺ 𝐵 → ¬ 𝐴 ≈ 𝐵)
13	12	adantr 480	. . . . . . 7 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ¬ 𝐴 ≈ 𝐵)
14		ssdif0 4316	. . . . . . . 8 ⊢ (𝐵 ⊆ ran 𝑓 ↔ (𝐵 ∖ ran 𝑓) = ∅)
15		simplr 768	. . . . . . . . . . 11 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) ∧ 𝐵 ⊆ ran 𝑓) → 𝑓:𝐴–1-1→𝐵)
16		f1f 6728	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴⟶𝐵)
17	16	frnd 6668	. . . . . . . . . . . . 13 ⊢ (𝑓:𝐴–1-1→𝐵 → ran 𝑓 ⊆ 𝐵)
18	15, 17	syl 17	. . . . . . . . . . . 12 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) ∧ 𝐵 ⊆ ran 𝑓) → ran 𝑓 ⊆ 𝐵)
19		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) ∧ 𝐵 ⊆ ran 𝑓) → 𝐵 ⊆ ran 𝑓)
20	18, 19	eqssd 3949	. . . . . . . . . . 11 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) ∧ 𝐵 ⊆ ran 𝑓) → ran 𝑓 = 𝐵)
21		dff1o5 6781	. . . . . . . . . . 11 ⊢ (𝑓:𝐴–1-1-onto→𝐵 ↔ (𝑓:𝐴–1-1→𝐵 ∧ ran 𝑓 = 𝐵))
22	15, 20, 21	sylanbrc 583	. . . . . . . . . 10 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) ∧ 𝐵 ⊆ ran 𝑓) → 𝑓:𝐴–1-1-onto→𝐵)
23		f1oen3g 8901	. . . . . . . . . 10 ⊢ ((𝑓 ∈ V ∧ 𝑓:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵)
24	4, 22, 23	sylancr 587	. . . . . . . . 9 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) ∧ 𝐵 ⊆ ran 𝑓) → 𝐴 ≈ 𝐵)
25	24	ex 412	. . . . . . . 8 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → (𝐵 ⊆ ran 𝑓 → 𝐴 ≈ 𝐵))
26	14, 25	biimtrrid 243	. . . . . . 7 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ((𝐵 ∖ ran 𝑓) = ∅ → 𝐴 ≈ 𝐵))
27	13, 26	mtod 198	. . . . . 6 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ¬ (𝐵 ∖ ran 𝑓) = ∅)
28		neq0 4302	. . . . . 6 ⊢ (¬ (𝐵 ∖ ran 𝑓) = ∅ ↔ ∃𝑤 𝑤 ∈ (𝐵 ∖ ran 𝑓))
29	27, 28	sylib 218	. . . . 5 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ∃𝑤 𝑤 ∈ (𝐵 ∖ ran 𝑓))
30		snssi 4762	. . . . . . 7 ⊢ (𝑤 ∈ (𝐵 ∖ ran 𝑓) → {𝑤} ⊆ (𝐵 ∖ ran 𝑓))
31		relsdom 8888	. . . . . . . . . . 11 ⊢ Rel ≺
32	31	brrelex1i 5678	. . . . . . . . . 10 ⊢ (𝐴 ≺ 𝐵 → 𝐴 ∈ V)
33	32	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → 𝐴 ∈ V)
34		vex 3442	. . . . . . . . 9 ⊢ 𝑤 ∈ V
35		en2sn 8976	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ 𝑤 ∈ V) → {𝐴} ≈ {𝑤})
36	33, 34, 35	sylancl 586	. . . . . . . 8 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → {𝐴} ≈ {𝑤})
37	31	brrelex2i 5679	. . . . . . . . . 10 ⊢ (𝐴 ≺ 𝐵 → 𝐵 ∈ V)
38	37	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → 𝐵 ∈ V)
39		difexg 5272	. . . . . . . . 9 ⊢ (𝐵 ∈ V → (𝐵 ∖ ran 𝑓) ∈ V)
40		snfi 8978	. . . . . . . . . . 11 ⊢ {𝑤} ∈ Fin
41		ssdomfi2 9119	. . . . . . . . . . 11 ⊢ (({𝑤} ∈ Fin ∧ (𝐵 ∖ ran 𝑓) ∈ V ∧ {𝑤} ⊆ (𝐵 ∖ ran 𝑓)) → {𝑤} ≼ (𝐵 ∖ ran 𝑓))
42	40, 41	mp3an1 1450	. . . . . . . . . 10 ⊢ (((𝐵 ∖ ran 𝑓) ∈ V ∧ {𝑤} ⊆ (𝐵 ∖ ran 𝑓)) → {𝑤} ≼ (𝐵 ∖ ran 𝑓))
43	42	ex 412	. . . . . . . . 9 ⊢ ((𝐵 ∖ ran 𝑓) ∈ V → ({𝑤} ⊆ (𝐵 ∖ ran 𝑓) → {𝑤} ≼ (𝐵 ∖ ran 𝑓)))
44	38, 39, 43	3syl 18	. . . . . . . 8 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ({𝑤} ⊆ (𝐵 ∖ ran 𝑓) → {𝑤} ≼ (𝐵 ∖ ran 𝑓)))
45		endom 8914	. . . . . . . . 9 ⊢ ({𝐴} ≈ {𝑤} → {𝐴} ≼ {𝑤})
46		domtrfi 9115	. . . . . . . . . 10 ⊢ (({𝑤} ∈ Fin ∧ {𝐴} ≼ {𝑤} ∧ {𝑤} ≼ (𝐵 ∖ ran 𝑓)) → {𝐴} ≼ (𝐵 ∖ ran 𝑓))
47	40, 46	mp3an1 1450	. . . . . . . . 9 ⊢ (({𝐴} ≼ {𝑤} ∧ {𝑤} ≼ (𝐵 ∖ ran 𝑓)) → {𝐴} ≼ (𝐵 ∖ ran 𝑓))
48	45, 47	sylan 580	. . . . . . . 8 ⊢ (({𝐴} ≈ {𝑤} ∧ {𝑤} ≼ (𝐵 ∖ ran 𝑓)) → {𝐴} ≼ (𝐵 ∖ ran 𝑓))
49	36, 44, 48	syl6an 684	. . . . . . 7 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ({𝑤} ⊆ (𝐵 ∖ ran 𝑓) → {𝐴} ≼ (𝐵 ∖ ran 𝑓)))
50	30, 49	syl5 34	. . . . . 6 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → (𝑤 ∈ (𝐵 ∖ ran 𝑓) → {𝐴} ≼ (𝐵 ∖ ran 𝑓)))
51	50	exlimdv 1934	. . . . 5 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → (∃𝑤 𝑤 ∈ (𝐵 ∖ ran 𝑓) → {𝐴} ≼ (𝐵 ∖ ran 𝑓)))
52	29, 51	mpd 15	. . . 4 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → {𝐴} ≼ (𝐵 ∖ ran 𝑓))
53		disjdif 4422	. . . . 5 ⊢ (ran 𝑓 ∩ (𝐵 ∖ ran 𝑓)) = ∅
54	53	a1i 11	. . . 4 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → (ran 𝑓 ∩ (𝐵 ∖ ran 𝑓)) = ∅)
55		undom 8991	. . . 4 ⊢ (((𝐴 ≼ ran 𝑓 ∧ {𝐴} ≼ (𝐵 ∖ ran 𝑓)) ∧ (ran 𝑓 ∩ (𝐵 ∖ ran 𝑓)) = ∅) → (𝐴 ∪ {𝐴}) ≼ (ran 𝑓 ∪ (𝐵 ∖ ran 𝑓)))
56	11, 52, 54, 55	syl21anc 837	. . 3 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → (𝐴 ∪ {𝐴}) ≼ (ran 𝑓 ∪ (𝐵 ∖ ran 𝑓)))
57		df-suc 6321	. . . 4 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
58	57	a1i 11	. . 3 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → suc 𝐴 = (𝐴 ∪ {𝐴}))
59		undif2 4427	. . . 4 ⊢ (ran 𝑓 ∪ (𝐵 ∖ ran 𝑓)) = (ran 𝑓 ∪ 𝐵)
60	17	adantl 481	. . . . 5 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ran 𝑓 ⊆ 𝐵)
61		ssequn1 4136	. . . . 5 ⊢ (ran 𝑓 ⊆ 𝐵 ↔ (ran 𝑓 ∪ 𝐵) = 𝐵)
62	60, 61	sylib 218	. . . 4 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → (ran 𝑓 ∪ 𝐵) = 𝐵)
63	59, 62	eqtr2id 2782	. . 3 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → 𝐵 = (ran 𝑓 ∪ (𝐵 ∖ ran 𝑓)))
64	56, 58, 63	3brtr4d 5128	. 2 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → suc 𝐴 ≼ 𝐵)
65	3, 64	exlimddv 1936	1 ⊢ (𝐴 ≺ 𝐵 → suc 𝐴 ≼ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  Vcvv 3438   ∖ cdif 3896   ∪ cun 3897   ∩ cin 3898   ⊆ wss 3899  ∅c0 4283  {csn 4578   class class class wbr 5096  ran crn 5623  suc csuc 6317  –1-1→wf1 6487  –1-1-onto→wf1o 6489   ≈ cen 8878   ≼ cdom 8879   ≺ csdm 8880  Fincfn 8881

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-om 7807  df-1o 8395  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885

This theorem is referenced by:  sucdom  9142  card2inf  9458