Theorem sucdom2 9128

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 5295. (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 8918	. . 3 ⊢ (𝐴 ≺ 𝐵 → 𝐴 ≼ 𝐵)
2		brdomi 8897	. . 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 3435	. . . . 5 ⊢ 𝑓 ∈ V
5	4	rnex 7851	. . . . 5 ⊢ ran 𝑓 ∈ V
6		f1f1orn 6779	. . . . . . 7 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴–1-1-onto→ran 𝑓)
7	6	adantl 482	. . . . . 6 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → 𝑓:𝐴–1-1-onto→ran 𝑓)
8		f1of1 6767	. . . . . 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 8905	. . . . 5 ⊢ ((𝑓 ∈ V ∧ ran 𝑓 ∈ V ∧ 𝑓:𝐴–1-1→ran 𝑓) → 𝐴 ≼ ran 𝑓)
11	4, 5, 9, 10	mp3an12i 1473	. . . 4 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → 𝐴 ≼ ran 𝑓)
12		sdomnen 8919	. . . . . . . 8 ⊢ (𝐴 ≺ 𝐵 → ¬ 𝐴 ≈ 𝐵)
13	12	adantr 481	. . . . . . 7 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ¬ 𝐴 ≈ 𝐵)
14		ssdif0 4295	. . . . . . . 8 ⊢ (𝐵 ⊆ ran 𝑓 ↔ (𝐵 ∖ ran 𝑓) = ∅)
15		simplr 774	. . . . . . . . . . 11 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) ∧ 𝐵 ⊆ ran 𝑓) → 𝑓:𝐴–1-1→𝐵)
16		f1f 6724	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴⟶𝐵)
17	16	frnd 6664	. . . . . . . . . . . . 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 485	. . . . . . . . . . . 12 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) ∧ 𝐵 ⊆ ran 𝑓) → 𝐵 ⊆ ran 𝑓)
20	18, 19	eqssd 3932	. . . . . . . . . . 11 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) ∧ 𝐵 ⊆ ran 𝑓) → ran 𝑓 = 𝐵)
21		dff1o5 6777	. . . . . . . . . . 11 ⊢ (𝑓:𝐴–1-1-onto→𝐵 ↔ (𝑓:𝐴–1-1→𝐵 ∧ ran 𝑓 = 𝐵))
22	15, 20, 21	sylanbrc 589	. . . . . . . . . 10 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) ∧ 𝐵 ⊆ ran 𝑓) → 𝑓:𝐴–1-1-onto→𝐵)
23		f1oen3g 8904	. . . . . . . . . 10 ⊢ ((𝑓 ∈ V ∧ 𝑓:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵)
24	4, 22, 23	sylancr 593	. . . . . . . . 9 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) ∧ 𝐵 ⊆ ran 𝑓) → 𝐴 ≈ 𝐵)
25	24	ex 413	. . . . . . . 8 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → (𝐵 ⊆ ran 𝑓 → 𝐴 ≈ 𝐵))
26	14, 25	biimtrrid 244	. . . . . . 7 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ((𝐵 ∖ ran 𝑓) = ∅ → 𝐴 ≈ 𝐵))
27	13, 26	mtod 199	. . . . . 6 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ¬ (𝐵 ∖ ran 𝑓) = ∅)
28		neq0 4281	. . . . . 6 ⊢ (¬ (𝐵 ∖ ran 𝑓) = ∅ ↔ ∃𝑤 𝑤 ∈ (𝐵 ∖ ran 𝑓))
29	27, 28	sylib 219	. . . . 5 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ∃𝑤 𝑤 ∈ (𝐵 ∖ ran 𝑓))
30		snssi 4718	. . . . . . 7 ⊢ (𝑤 ∈ (𝐵 ∖ ran 𝑓) → {𝑤} ⊆ (𝐵 ∖ ran 𝑓))
31		relsdom 8891	. . . . . . . . . . 11 ⊢ Rel ≺
32	31	brrelex1i 5675	. . . . . . . . . 10 ⊢ (𝐴 ≺ 𝐵 → 𝐴 ∈ V)
33	32	adantr 481	. . . . . . . . 9 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → 𝐴 ∈ V)
34		vex 3435	. . . . . . . . 9 ⊢ 𝑤 ∈ V
35		en2sn 8979	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ 𝑤 ∈ V) → {𝐴} ≈ {𝑤})
36	33, 34, 35	sylancl 592	. . . . . . . 8 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → {𝐴} ≈ {𝑤})
37	31	brrelex2i 5676	. . . . . . . . . 10 ⊢ (𝐴 ≺ 𝐵 → 𝐵 ∈ V)
38	37	adantr 481	. . . . . . . . 9 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → 𝐵 ∈ V)
39		difexg 5258	. . . . . . . . 9 ⊢ (𝐵 ∈ V → (𝐵 ∖ ran 𝑓) ∈ V)
40		snfi 8981	. . . . . . . . . . 11 ⊢ {𝑤} ∈ Fin
41		ssdomfi2 9122	. . . . . . . . . . 11 ⊢ (({𝑤} ∈ Fin ∧ (𝐵 ∖ ran 𝑓) ∈ V ∧ {𝑤} ⊆ (𝐵 ∖ ran 𝑓)) → {𝑤} ≼ (𝐵 ∖ ran 𝑓))
42	40, 41	mp3an1 1456	. . . . . . . . . 10 ⊢ (((𝐵 ∖ ran 𝑓) ∈ V ∧ {𝑤} ⊆ (𝐵 ∖ ran 𝑓)) → {𝑤} ≼ (𝐵 ∖ ran 𝑓))
43	42	ex 413	. . . . . . . . 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 8917	. . . . . . . . 9 ⊢ ({𝐴} ≈ {𝑤} → {𝐴} ≼ {𝑤})
46		domtrfi 9118	. . . . . . . . . 10 ⊢ (({𝑤} ∈ Fin ∧ {𝐴} ≼ {𝑤} ∧ {𝑤} ≼ (𝐵 ∖ ran 𝑓)) → {𝐴} ≼ (𝐵 ∖ ran 𝑓))
47	40, 46	mp3an1 1456	. . . . . . . . 9 ⊢ (({𝐴} ≼ {𝑤} ∧ {𝑤} ≼ (𝐵 ∖ ran 𝑓)) → {𝐴} ≼ (𝐵 ∖ ran 𝑓))
48	45, 47	sylan 586	. . . . . . . 8 ⊢ (({𝐴} ≈ {𝑤} ∧ {𝑤} ≼ (𝐵 ∖ ran 𝑓)) → {𝐴} ≼ (𝐵 ∖ ran 𝑓))
49	36, 44, 48	syl6an 690	. . . . . . 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 1940	. . . . 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 4401	. . . . 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 8994	. . . 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 843	. . 3 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → (𝐴 ∪ {𝐴}) ≼ (ran 𝑓 ∪ (𝐵 ∖ ran 𝑓)))
57		df-suc 6317	. . . 4 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
58	57	a1i 11	. . 3 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → suc 𝐴 = (𝐴 ∪ {𝐴}))
59		undif2 4406	. . . 4 ⊢ (ran 𝑓 ∪ (𝐵 ∖ ran 𝑓)) = (ran 𝑓 ∪ 𝐵)
60	17	adantl 482	. . . . 5 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ran 𝑓 ⊆ 𝐵)
61		ssequn1 4116	. . . . 5 ⊢ (ran 𝑓 ⊆ 𝐵 ↔ (ran 𝑓 ∪ 𝐵) = 𝐵)
62	60, 61	sylib 219	. . . 4 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → (ran 𝑓 ∪ 𝐵) = 𝐵)
63	59, 62	eqtr2id 2787	. . 3 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → 𝐵 = (ran 𝑓 ∪ (𝐵 ∖ ran 𝑓)))
64	56, 58, 63	3brtr4d 5105	. 2 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → suc 𝐴 ≼ 𝐵)
65	3, 64	exlimddv 1942	1 ⊢ (𝐴 ≺ 𝐵 → suc 𝐴 ≼ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1547  ∃wex 1786   ∈ wcel 2119  Vcvv 3431   ∖ cdif 3880   ∪ cun 3881   ∩ cin 3882   ⊆ wss 3883  ∅c0 4262  {csn 4556   class class class wbr 5073  ran crn 5620  suc csuc 6313  –1-1→wf1 6483  –1-1-onto→wf1o 6485   ≈ cen 8881   ≼ cdom 8882   ≺ csdm 8883  Fincfn 8884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5219  ax-nul 5229  ax-pr 5363  ax-un 7679

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-br 5074  df-opab 5136  df-tr 5181  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-om 7808  df-1o 8396  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888

This theorem is referenced by:  sucdom  9145  card2inf  9461