Theorem sucdom2 8708

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.)

Assertion

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

Proof of Theorem sucdom2

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

Step	Hyp	Ref	Expression
1		sdomdom 8531	. . 3 ⊢ (𝐴 ≺ 𝐵 → 𝐴 ≼ 𝐵)
2		brdomi 8514	. . 3 ⊢ (𝐴 ≼ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵)
3	1, 2	syl 17	. 2 ⊢ (𝐴 ≺ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵)
4		relsdom 8510	. . . . . . 7 ⊢ Rel ≺
5	4	brrelex1i 5602	. . . . . 6 ⊢ (𝐴 ≺ 𝐵 → 𝐴 ∈ V)
6	5	adantr 483	. . . . 5 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → 𝐴 ∈ V)
7		vex 3497	. . . . . . 7 ⊢ 𝑓 ∈ V
8	7	rnex 7611	. . . . . 6 ⊢ ran 𝑓 ∈ V
9	8	a1i 11	. . . . 5 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ran 𝑓 ∈ V)
10		f1f1orn 6620	. . . . . . 7 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴–1-1-onto→ran 𝑓)
11	10	adantl 484	. . . . . 6 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → 𝑓:𝐴–1-1-onto→ran 𝑓)
12		f1of1 6608	. . . . . 6 ⊢ (𝑓:𝐴–1-1-onto→ran 𝑓 → 𝑓:𝐴–1-1→ran 𝑓)
13	11, 12	syl 17	. . . . 5 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → 𝑓:𝐴–1-1→ran 𝑓)
14		f1dom2g 8521	. . . . 5 ⊢ ((𝐴 ∈ V ∧ ran 𝑓 ∈ V ∧ 𝑓:𝐴–1-1→ran 𝑓) → 𝐴 ≼ ran 𝑓)
15	6, 9, 13, 14	syl3anc 1367	. . . 4 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → 𝐴 ≼ ran 𝑓)
16		sdomnen 8532	. . . . . . . 8 ⊢ (𝐴 ≺ 𝐵 → ¬ 𝐴 ≈ 𝐵)
17	16	adantr 483	. . . . . . 7 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ¬ 𝐴 ≈ 𝐵)
18		ssdif0 4322	. . . . . . . 8 ⊢ (𝐵 ⊆ ran 𝑓 ↔ (𝐵 ∖ ran 𝑓) = ∅)
19		simplr 767	. . . . . . . . . . 11 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) ∧ 𝐵 ⊆ ran 𝑓) → 𝑓:𝐴–1-1→𝐵)
20		f1f 6569	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴⟶𝐵)
21	20	frnd 6515	. . . . . . . . . . . . 13 ⊢ (𝑓:𝐴–1-1→𝐵 → ran 𝑓 ⊆ 𝐵)
22	19, 21	syl 17	. . . . . . . . . . . 12 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) ∧ 𝐵 ⊆ ran 𝑓) → ran 𝑓 ⊆ 𝐵)
23		simpr 487	. . . . . . . . . . . 12 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) ∧ 𝐵 ⊆ ran 𝑓) → 𝐵 ⊆ ran 𝑓)
24	22, 23	eqssd 3983	. . . . . . . . . . 11 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) ∧ 𝐵 ⊆ ran 𝑓) → ran 𝑓 = 𝐵)
25		dff1o5 6618	. . . . . . . . . . 11 ⊢ (𝑓:𝐴–1-1-onto→𝐵 ↔ (𝑓:𝐴–1-1→𝐵 ∧ ran 𝑓 = 𝐵))
26	19, 24, 25	sylanbrc 585	. . . . . . . . . 10 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) ∧ 𝐵 ⊆ ran 𝑓) → 𝑓:𝐴–1-1-onto→𝐵)
27		f1oen3g 8519	. . . . . . . . . 10 ⊢ ((𝑓 ∈ V ∧ 𝑓:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵)
28	7, 26, 27	sylancr 589	. . . . . . . . 9 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) ∧ 𝐵 ⊆ ran 𝑓) → 𝐴 ≈ 𝐵)
29	28	ex 415	. . . . . . . 8 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → (𝐵 ⊆ ran 𝑓 → 𝐴 ≈ 𝐵))
30	18, 29	syl5bir 245	. . . . . . 7 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ((𝐵 ∖ ran 𝑓) = ∅ → 𝐴 ≈ 𝐵))
31	17, 30	mtod 200	. . . . . 6 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ¬ (𝐵 ∖ ran 𝑓) = ∅)
32		neq0 4308	. . . . . 6 ⊢ (¬ (𝐵 ∖ ran 𝑓) = ∅ ↔ ∃𝑤 𝑤 ∈ (𝐵 ∖ ran 𝑓))
33	31, 32	sylib 220	. . . . 5 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ∃𝑤 𝑤 ∈ (𝐵 ∖ ran 𝑓))
34		snssi 4734	. . . . . . 7 ⊢ (𝑤 ∈ (𝐵 ∖ ran 𝑓) → {𝑤} ⊆ (𝐵 ∖ ran 𝑓))
35		vex 3497	. . . . . . . . 9 ⊢ 𝑤 ∈ V
36		en2sn 8587	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ 𝑤 ∈ V) → {𝐴} ≈ {𝑤})
37	6, 35, 36	sylancl 588	. . . . . . . 8 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → {𝐴} ≈ {𝑤})
38	4	brrelex2i 5603	. . . . . . . . . 10 ⊢ (𝐴 ≺ 𝐵 → 𝐵 ∈ V)
39	38	adantr 483	. . . . . . . . 9 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → 𝐵 ∈ V)
40		difexg 5223	. . . . . . . . 9 ⊢ (𝐵 ∈ V → (𝐵 ∖ ran 𝑓) ∈ V)
41		ssdomg 8549	. . . . . . . . 9 ⊢ ((𝐵 ∖ ran 𝑓) ∈ V → ({𝑤} ⊆ (𝐵 ∖ ran 𝑓) → {𝑤} ≼ (𝐵 ∖ ran 𝑓)))
42	39, 40, 41	3syl 18	. . . . . . . 8 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ({𝑤} ⊆ (𝐵 ∖ ran 𝑓) → {𝑤} ≼ (𝐵 ∖ ran 𝑓)))
43		endomtr 8561	. . . . . . . 8 ⊢ (({𝐴} ≈ {𝑤} ∧ {𝑤} ≼ (𝐵 ∖ ran 𝑓)) → {𝐴} ≼ (𝐵 ∖ ran 𝑓))
44	37, 42, 43	syl6an 682	. . . . . . 7 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ({𝑤} ⊆ (𝐵 ∖ ran 𝑓) → {𝐴} ≼ (𝐵 ∖ ran 𝑓)))
45	34, 44	syl5 34	. . . . . 6 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → (𝑤 ∈ (𝐵 ∖ ran 𝑓) → {𝐴} ≼ (𝐵 ∖ ran 𝑓)))
46	45	exlimdv 1930	. . . . 5 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → (∃𝑤 𝑤 ∈ (𝐵 ∖ ran 𝑓) → {𝐴} ≼ (𝐵 ∖ ran 𝑓)))
47	33, 46	mpd 15	. . . 4 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → {𝐴} ≼ (𝐵 ∖ ran 𝑓))
48		disjdif 4420	. . . . 5 ⊢ (ran 𝑓 ∩ (𝐵 ∖ ran 𝑓)) = ∅
49	48	a1i 11	. . . 4 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → (ran 𝑓 ∩ (𝐵 ∖ ran 𝑓)) = ∅)
50		undom 8599	. . . 4 ⊢ (((𝐴 ≼ ran 𝑓 ∧ {𝐴} ≼ (𝐵 ∖ ran 𝑓)) ∧ (ran 𝑓 ∩ (𝐵 ∖ ran 𝑓)) = ∅) → (𝐴 ∪ {𝐴}) ≼ (ran 𝑓 ∪ (𝐵 ∖ ran 𝑓)))
51	15, 47, 49, 50	syl21anc 835	. . 3 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → (𝐴 ∪ {𝐴}) ≼ (ran 𝑓 ∪ (𝐵 ∖ ran 𝑓)))
52		df-suc 6191	. . . 4 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
53	52	a1i 11	. . 3 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → suc 𝐴 = (𝐴 ∪ {𝐴}))
54		undif2 4424	. . . 4 ⊢ (ran 𝑓 ∪ (𝐵 ∖ ran 𝑓)) = (ran 𝑓 ∪ 𝐵)
55	21	adantl 484	. . . . 5 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ran 𝑓 ⊆ 𝐵)
56		ssequn1 4155	. . . . 5 ⊢ (ran 𝑓 ⊆ 𝐵 ↔ (ran 𝑓 ∪ 𝐵) = 𝐵)
57	55, 56	sylib 220	. . . 4 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → (ran 𝑓 ∪ 𝐵) = 𝐵)
58	54, 57	syl5req 2869	. . 3 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → 𝐵 = (ran 𝑓 ∪ (𝐵 ∖ ran 𝑓)))
59	51, 53, 58	3brtr4d 5090	. 2 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → suc 𝐴 ≼ 𝐵)
60	3, 59	exlimddv 1932	1 ⊢ (𝐴 ≺ 𝐵 → suc 𝐴 ≼ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1533  ∃wex 1776   ∈ wcel 2110  Vcvv 3494   ∖ cdif 3932   ∪ cun 3933   ∩ cin 3934   ⊆ wss 3935  ∅c0 4290  {csn 4560   class class class wbr 5058  ran crn 5550  suc csuc 6187  –1-1→wf1 6346  –1-1-onto→wf1o 6348   ≈ cen 8500   ≼ cdom 8501   ≺ csdm 8502

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-suc 6191  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-1o 8096  df-er 8283  df-en 8504  df-dom 8505  df-sdom 8506

This theorem is referenced by:  sucdom  8709  card2inf  9013