Theorem sucdom2 9241

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 5371. (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 9019	. . 3 ⊢ (𝐴 ≺ 𝐵 → 𝐴 ≼ 𝐵)
2		brdomi 8998	. . 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 3482	. . . . 5 ⊢ 𝑓 ∈ V
5	4	rnex 7933	. . . . 5 ⊢ ran 𝑓 ∈ V
6		f1f1orn 6860	. . . . . . 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 6848	. . . . . 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 9007	. . . . 5 ⊢ ((𝑓 ∈ V ∧ ran 𝑓 ∈ V ∧ 𝑓:𝐴–1-1→ran 𝑓) → 𝐴 ≼ ran 𝑓)
11	4, 5, 9, 10	mp3an12i 1464	. . . 4 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → 𝐴 ≼ ran 𝑓)
12		sdomnen 9020	. . . . . . . 8 ⊢ (𝐴 ≺ 𝐵 → ¬ 𝐴 ≈ 𝐵)
13	12	adantr 480	. . . . . . 7 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ¬ 𝐴 ≈ 𝐵)
14		ssdif0 4372	. . . . . . . 8 ⊢ (𝐵 ⊆ ran 𝑓 ↔ (𝐵 ∖ ran 𝑓) = ∅)
15		simplr 769	. . . . . . . . . . 11 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) ∧ 𝐵 ⊆ ran 𝑓) → 𝑓:𝐴–1-1→𝐵)
16		f1f 6805	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴⟶𝐵)
17	16	frnd 6745	. . . . . . . . . . . . 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 4013	. . . . . . . . . . 11 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) ∧ 𝐵 ⊆ ran 𝑓) → ran 𝑓 = 𝐵)
21		dff1o5 6858	. . . . . . . . . . 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 9006	. . . . . . . . . 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 4358	. . . . . 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 4813	. . . . . . 7 ⊢ (𝑤 ∈ (𝐵 ∖ ran 𝑓) → {𝑤} ⊆ (𝐵 ∖ ran 𝑓))
31		relsdom 8991	. . . . . . . . . . 11 ⊢ Rel ≺
32	31	brrelex1i 5745	. . . . . . . . . 10 ⊢ (𝐴 ≺ 𝐵 → 𝐴 ∈ V)
33	32	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → 𝐴 ∈ V)
34		vex 3482	. . . . . . . . 9 ⊢ 𝑤 ∈ V
35		en2sn 9080	. . . . . . . . 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 5746	. . . . . . . . . 10 ⊢ (𝐴 ≺ 𝐵 → 𝐵 ∈ V)
38	37	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → 𝐵 ∈ V)
39		difexg 5335	. . . . . . . . 9 ⊢ (𝐵 ∈ V → (𝐵 ∖ ran 𝑓) ∈ V)
40		snfi 9082	. . . . . . . . . . 11 ⊢ {𝑤} ∈ Fin
41		ssdomfi2 9235	. . . . . . . . . . 11 ⊢ (({𝑤} ∈ Fin ∧ (𝐵 ∖ ran 𝑓) ∈ V ∧ {𝑤} ⊆ (𝐵 ∖ ran 𝑓)) → {𝑤} ≼ (𝐵 ∖ ran 𝑓))
42	40, 41	mp3an1 1447	. . . . . . . . . 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 9018	. . . . . . . . 9 ⊢ ({𝐴} ≈ {𝑤} → {𝐴} ≼ {𝑤})
46		domtrfi 9231	. . . . . . . . . 10 ⊢ (({𝑤} ∈ Fin ∧ {𝐴} ≼ {𝑤} ∧ {𝑤} ≼ (𝐵 ∖ ran 𝑓)) → {𝐴} ≼ (𝐵 ∖ ran 𝑓))
47	40, 46	mp3an1 1447	. . . . . . . . 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 1931	. . . . 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 4478	. . . . 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 9098	. . . 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 838	. . 3 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → (𝐴 ∪ {𝐴}) ≼ (ran 𝑓 ∪ (𝐵 ∖ ran 𝑓)))
57		df-suc 6392	. . . 4 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
58	57	a1i 11	. . 3 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → suc 𝐴 = (𝐴 ∪ {𝐴}))
59		undif2 4483	. . . 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 4196	. . . . 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 2788	. . 3 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → 𝐵 = (ran 𝑓 ∪ (𝐵 ∖ ran 𝑓)))
64	56, 58, 63	3brtr4d 5180	. 2 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → suc 𝐴 ≼ 𝐵)
65	3, 64	exlimddv 1933	1 ⊢ (𝐴 ≺ 𝐵 → suc 𝐴 ≼ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1537  ∃wex 1776   ∈ wcel 2106  Vcvv 3478   ∖ cdif 3960   ∪ cun 3961   ∩ cin 3962   ⊆ wss 3963  ∅c0 4339  {csn 4631   class class class wbr 5148  ran crn 5690  suc csuc 6388  –1-1→wf1 6560  –1-1-onto→wf1o 6562   ≈ cen 8981   ≼ cdom 8982   ≺ csdm 8983  Fincfn 8984

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-1o 8505  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988

This theorem is referenced by:  sucdom  9269  sucdomOLD  9270  card2inf  9593