Theorem sucdom2OLD 9121

Description: Obsolete version of sucdom2 9241 as of 4-Dec-2024. (Contributed by Mario Carneiro, 12-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
sucdom2OLD	⊢ (𝐴 ≺ 𝐵 → suc 𝐴 ≼ 𝐵)

Proof of Theorem sucdom2OLD

Dummy variables 𝑤 𝑓 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		relsdom 8991	. . . . . . 7 ⊢ Rel ≺
5	4	brrelex1i 5745	. . . . . 6 ⊢ (𝐴 ≺ 𝐵 → 𝐴 ∈ V)
6	5	adantr 480	. . . . 5 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → 𝐴 ∈ V)
7		vex 3482	. . . . . . 7 ⊢ 𝑓 ∈ V
8	7	rnex 7933	. . . . . 6 ⊢ ran 𝑓 ∈ V
9	8	a1i 11	. . . . 5 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ran 𝑓 ∈ V)
10		f1f1orn 6860	. . . . . . 7 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴–1-1-onto→ran 𝑓)
11	10	adantl 481	. . . . . 6 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → 𝑓:𝐴–1-1-onto→ran 𝑓)
12		f1of1 6848	. . . . . 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 9009	. . . . 5 ⊢ ((𝐴 ∈ V ∧ ran 𝑓 ∈ V ∧ 𝑓:𝐴–1-1→ran 𝑓) → 𝐴 ≼ ran 𝑓)
15	6, 9, 13, 14	syl3anc 1370	. . . 4 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → 𝐴 ≼ ran 𝑓)
16		sdomnen 9020	. . . . . . . 8 ⊢ (𝐴 ≺ 𝐵 → ¬ 𝐴 ≈ 𝐵)
17	16	adantr 480	. . . . . . 7 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ¬ 𝐴 ≈ 𝐵)
18		ssdif0 4372	. . . . . . . 8 ⊢ (𝐵 ⊆ ran 𝑓 ↔ (𝐵 ∖ ran 𝑓) = ∅)
19		simplr 769	. . . . . . . . . . 11 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) ∧ 𝐵 ⊆ ran 𝑓) → 𝑓:𝐴–1-1→𝐵)
20		f1f 6805	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐴–1-1→𝐵 → 𝑓:𝐴⟶𝐵)
21	20	frnd 6745	. . . . . . . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) ∧ 𝐵 ⊆ ran 𝑓) → 𝐵 ⊆ ran 𝑓)
24	22, 23	eqssd 4013	. . . . . . . . . . 11 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) ∧ 𝐵 ⊆ ran 𝑓) → ran 𝑓 = 𝐵)
25		dff1o5 6858	. . . . . . . . . . 11 ⊢ (𝑓:𝐴–1-1-onto→𝐵 ↔ (𝑓:𝐴–1-1→𝐵 ∧ ran 𝑓 = 𝐵))
26	19, 24, 25	sylanbrc 583	. . . . . . . . . 10 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) ∧ 𝐵 ⊆ ran 𝑓) → 𝑓:𝐴–1-1-onto→𝐵)
27		f1oen3g 9006	. . . . . . . . . 10 ⊢ ((𝑓 ∈ V ∧ 𝑓:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵)
28	7, 26, 27	sylancr 587	. . . . . . . . 9 ⊢ (((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) ∧ 𝐵 ⊆ ran 𝑓) → 𝐴 ≈ 𝐵)
29	28	ex 412	. . . . . . . 8 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → (𝐵 ⊆ ran 𝑓 → 𝐴 ≈ 𝐵))
30	18, 29	biimtrrid 243	. . . . . . 7 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ((𝐵 ∖ ran 𝑓) = ∅ → 𝐴 ≈ 𝐵))
31	17, 30	mtod 198	. . . . . 6 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ¬ (𝐵 ∖ ran 𝑓) = ∅)
32		neq0 4358	. . . . . 6 ⊢ (¬ (𝐵 ∖ ran 𝑓) = ∅ ↔ ∃𝑤 𝑤 ∈ (𝐵 ∖ ran 𝑓))
33	31, 32	sylib 218	. . . . 5 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ∃𝑤 𝑤 ∈ (𝐵 ∖ ran 𝑓))
34		snssi 4813	. . . . . . 7 ⊢ (𝑤 ∈ (𝐵 ∖ ran 𝑓) → {𝑤} ⊆ (𝐵 ∖ ran 𝑓))
35		vex 3482	. . . . . . . . 9 ⊢ 𝑤 ∈ V
36		en2sn 9080	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ 𝑤 ∈ V) → {𝐴} ≈ {𝑤})
37	6, 35, 36	sylancl 586	. . . . . . . 8 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → {𝐴} ≈ {𝑤})
38	4	brrelex2i 5746	. . . . . . . . . 10 ⊢ (𝐴 ≺ 𝐵 → 𝐵 ∈ V)
39	38	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → 𝐵 ∈ V)
40		difexg 5335	. . . . . . . . 9 ⊢ (𝐵 ∈ V → (𝐵 ∖ ran 𝑓) ∈ V)
41		ssdomg 9039	. . . . . . . . 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 9051	. . . . . . . 8 ⊢ (({𝐴} ≈ {𝑤} ∧ {𝑤} ≼ (𝐵 ∖ ran 𝑓)) → {𝐴} ≼ (𝐵 ∖ ran 𝑓))
44	37, 42, 43	syl6an 684	. . . . . . 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 1931	. . . . 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 4478	. . . . 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 9098	. . . 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 838	. . 3 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → (𝐴 ∪ {𝐴}) ≼ (ran 𝑓 ∪ (𝐵 ∖ ran 𝑓)))
52		df-suc 6392	. . . 4 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
53	52	a1i 11	. . 3 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → suc 𝐴 = (𝐴 ∪ {𝐴}))
54		undif2 4483	. . . 4 ⊢ (ran 𝑓 ∪ (𝐵 ∖ ran 𝑓)) = (ran 𝑓 ∪ 𝐵)
55	21	adantl 481	. . . . 5 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → ran 𝑓 ⊆ 𝐵)
56		ssequn1 4196	. . . . 5 ⊢ (ran 𝑓 ⊆ 𝐵 ↔ (ran 𝑓 ∪ 𝐵) = 𝐵)
57	55, 56	sylib 218	. . . 4 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → (ran 𝑓 ∪ 𝐵) = 𝐵)
58	54, 57	eqtr2id 2788	. . 3 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → 𝐵 = (ran 𝑓 ∪ (𝐵 ∖ ran 𝑓)))
59	51, 53, 58	3brtr4d 5180	. 2 ⊢ ((𝐴 ≺ 𝐵 ∧ 𝑓:𝐴–1-1→𝐵) → suc 𝐴 ≼ 𝐵)
60	3, 59	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

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-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  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-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  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-id 5583  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-suc 6392  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-en 8985  df-dom 8986  df-sdom 8987

This theorem is referenced by: (None)