Theorem unfilem1 4560

Description: Lemma for proving that the union of two finite sets is finite.

Hypotheses

Ref	Expression
unfilem1.1	⊢ A ∈ ω
unfilem1.2	⊢ B ∈ ω
unfilem1.3	⊢ F = {⟨x, y⟩∣(x ∈ B ⋀ y = (A +o x))}

Assertion

Ref	Expression
unfilem1	⊢ ran F = ((A +o B) ∖ A)

Distinct variable groups:   x,y,A   x,B,y

Proof of Theorem unfilem1

Step	Hyp	Ref	Expression
1		rnopab 3359	. 2 ⊢ ran {⟨x, y⟩∣(x ∈ B ⋀ y = (A +o x))} = {y∣∃x(x ∈ B ⋀ y = (A +o x))}
2		unfilem1.3	. . 3 ⊢ F = {⟨x, y⟩∣(x ∈ B ⋀ y = (A +o x))}
3	2	rneqi 3346	. 2 ⊢ ran F = ran {⟨x, y⟩∣(x ∈ B ⋀ y = (A +o x))}
4		eldif 2060	. . . 4 ⊢ (y ∈ ((A +o B) ∖ A) ↔ (y ∈ (A +o B) ⋀ ¬ y ∈ A))
5		unfilem1.1	. . . . . . . . . 10 ⊢ A ∈ ω
6		unfilem1.2	. . . . . . . . . 10 ⊢ B ∈ ω
7		nnacl 4235	. . . . . . . . . 10 ⊢ ((A ∈ ω ⋀ B ∈ ω) → (A +o B) ∈ ω)
8	5, 6, 7	mp2an 699	. . . . . . . . 9 ⊢ (A +o B) ∈ ω
9		elnn 3148	. . . . . . . . 9 ⊢ ((y ∈ (A +o B) ⋀ (A +o B) ∈ ω) → y ∈ ω)
10	8, 9	mpan2 698	. . . . . . . 8 ⊢ (y ∈ (A +o B) → y ∈ ω)
11		ordtri1 2986	. . . . . . . . . . . 12 ⊢ ((Ord A ⋀ Ord y) → (A ⊆ y ↔ ¬ y ∈ A))
12		nnord 3146	. . . . . . . . . . . 12 ⊢ (A ∈ ω → Ord A)
13		nnord 3146	. . . . . . . . . . . 12 ⊢ (y ∈ ω → Ord y)
14	11, 12, 13	syl2an 456	. . . . . . . . . . 11 ⊢ ((A ∈ ω ⋀ y ∈ ω) → (A ⊆ y ↔ ¬ y ∈ A))
15		nnawordex 4256	. . . . . . . . . . 11 ⊢ ((A ∈ ω ⋀ y ∈ ω) → (A ⊆ y ↔ ∃x ∈ ω (A +o x) = y))
16	14, 15	bitr3d 532	. . . . . . . . . 10 ⊢ ((A ∈ ω ⋀ y ∈ ω) → (¬ y ∈ A ↔ ∃x ∈ ω (A +o x) = y))
17	5, 16	mpan 697	. . . . . . . . 9 ⊢ (y ∈ ω → (¬ y ∈ A ↔ ∃x ∈ ω (A +o x) = y))
18		df-rex 1653	. . . . . . . . 9 ⊢ (∃x ∈ ω (A +o x) = y ↔ ∃x(x ∈ ω ⋀ (A +o x) = y))
19	17, 18	syl6bb 538	. . . . . . . 8 ⊢ (y ∈ ω → (¬ y ∈ A ↔ ∃x(x ∈ ω ⋀ (A +o x) = y)))
20	10, 19	syl 10	. . . . . . 7 ⊢ (y ∈ (A +o B) → (¬ y ∈ A ↔ ∃x(x ∈ ω ⋀ (A +o x) = y)))
21		nnaord 4241	. . . . . . . . . . . 12 ⊢ ((x ∈ ω ⋀ B ∈ ω ⋀ A ∈ ω) → (x ∈ B ↔ (A +o x) ∈ (A +o B)))
22	6, 5, 21	mp3an23 910	. . . . . . . . . . 11 ⊢ (x ∈ ω → (x ∈ B ↔ (A +o x) ∈ (A +o B)))
23		eleq1 1537	. . . . . . . . . . 11 ⊢ ((A +o x) = y → ((A +o x) ∈ (A +o B) ↔ y ∈ (A +o B)))
24	22, 23	sylan9bb 542	. . . . . . . . . 10 ⊢ ((x ∈ ω ⋀ (A +o x) = y) → (x ∈ B ↔ y ∈ (A +o B)))
25	24	biimprcd 156	. . . . . . . . 9 ⊢ (y ∈ (A +o B) → ((x ∈ ω ⋀ (A +o x) = y) → x ∈ B))
26		eqcom 1480	. . . . . . . . . . . 12 ⊢ ((A +o x) = y ↔ y = (A +o x))
27	26	biimp 151	. . . . . . . . . . 11 ⊢ ((A +o x) = y → y = (A +o x))
28	27	adantl 390	. . . . . . . . . 10 ⊢ ((x ∈ ω ⋀ (A +o x) = y) → y = (A +o x))
29	28	a1i 8	. . . . . . . . 9 ⊢ (y ∈ (A +o B) → ((x ∈ ω ⋀ (A +o x) = y) → y = (A +o x)))
30	25, 29	jcad 602	. . . . . . . 8 ⊢ (y ∈ (A +o B) → ((x ∈ ω ⋀ (A +o x) = y) → (x ∈ B ⋀ y = (A +o x))))
31	30	19.22dv 1292	. . . . . . 7 ⊢ (y ∈ (A +o B) → (∃x(x ∈ ω ⋀ (A +o x) = y) → ∃x(x ∈ B ⋀ y = (A +o x))))
32	20, 31	sylbid 203	. . . . . 6 ⊢ (y ∈ (A +o B) → (¬ y ∈ A → ∃x(x ∈ B ⋀ y = (A +o x))))
33	32	imp 350	. . . . 5 ⊢ ((y ∈ (A +o B) ⋀ ¬ y ∈ A) → ∃x(x ∈ B ⋀ y = (A +o x)))
34		eleq1 1537	. . . . . . . . 9 ⊢ (y = (A +o x) → (y ∈ (A +o B) ↔ (A +o x) ∈ (A +o B)))
35		eleq1 1537	. . . . . . . . . 10 ⊢ (y = (A +o x) → (y ∈ A ↔ (A +o x) ∈ A))
36	35	negbid 613	. . . . . . . . 9 ⊢ (y = (A +o x) → (¬ y ∈ A ↔ ¬ (A +o x) ∈ A))
37	34, 36	anbi12d 630	. . . . . . . 8 ⊢ (y = (A +o x) → ((y ∈ (A +o B) ⋀ ¬ y ∈ A) ↔ ((A +o x) ∈ (A +o B) ⋀ ¬ (A +o x) ∈ A)))
38	37	biimparc 421	. . . . . . 7 ⊢ ((((A +o x) ∈ (A +o B) ⋀ ¬ (A +o x) ∈ A) ⋀ y = (A +o x)) → (y ∈ (A +o B) ⋀ ¬ y ∈ A))
39		elnn 3148	. . . . . . . . . . 11 ⊢ ((x ∈ B ⋀ B ∈ ω) → x ∈ ω)
40	6, 39	mpan2 698	. . . . . . . . . 10 ⊢ (x ∈ B → x ∈ ω)
41	40, 22	syl 10	. . . . . . . . 9 ⊢ (x ∈ B → (x ∈ B ↔ (A +o x) ∈ (A +o B)))
42	41	ibi 594	. . . . . . . 8 ⊢ (x ∈ B → (A +o x) ∈ (A +o B))
43		nnaword1 4250	. . . . . . . . . . 11 ⊢ ((A ∈ ω ⋀ x ∈ ω) → A ⊆ (A +o x))
44		nnacl 4235	. . . . . . . . . . . 12 ⊢ ((A ∈ ω ⋀ x ∈ ω) → (A +o x) ∈ ω)
45		nnord 3146	. . . . . . . . . . . 12 ⊢ ((A +o x) ∈ ω → Ord (A +o x))
46	5, 12	ax-mp 7	. . . . . . . . . . . . 13 ⊢ Ord A
47		ordtri1 2986	. . . . . . . . . . . . 13 ⊢ ((Ord A ⋀ Ord (A +o x)) → (A ⊆ (A +o x) ↔ ¬ (A +o x) ∈ A))
48	46, 47	mpan 697	. . . . . . . . . . . 12 ⊢ (Ord (A +o x) → (A ⊆ (A +o x) ↔ ¬ (A +o x) ∈ A))
49	44, 45, 48	3syl 20	. . . . . . . . . . 11 ⊢ ((A ∈ ω ⋀ x ∈ ω) → (A ⊆ (A +o x) ↔ ¬ (A +o x) ∈ A))
50	43, 49	mpbid 195	. . . . . . . . . 10 ⊢ ((A ∈ ω ⋀ x ∈ ω) → ¬ (A +o x) ∈ A)
51	5, 50	mpan 697	. . . . . . . . 9 ⊢ (x ∈ ω → ¬ (A +o x) ∈ A)
52	40, 51	syl 10	. . . . . . . 8 ⊢ (x ∈ B → ¬ (A +o x) ∈ A)
53	42, 52	jca 288	. . . . . . 7 ⊢ (x ∈ B → ((A +o x) ∈ (A +o B) ⋀ ¬ (A +o x) ∈ A))
54	38, 53	sylan 450	. . . . . 6 ⊢ ((x ∈ B ⋀ y = (A +o x)) → (y ∈ (A +o B) ⋀ ¬ y ∈ A))
55	54	19.23aiv 1297	. . . . 5 ⊢ (∃x(x ∈ B ⋀ y = (A +o x)) → (y ∈ (A +o B) ⋀ ¬ y ∈ A))
56	33, 55	impbi 157	. . . 4 ⊢ ((y ∈ (A +o B) ⋀ ¬ y ∈ A) ↔ ∃x(x ∈ B ⋀ y = (A +o x)))
57	4, 56	bitr 173	. . 3 ⊢ (y ∈ ((A +o B) ∖ A) ↔ ∃x(x ∈ B ⋀ y = (A +o x)))
58	57	abbi2i 1577	. 2 ⊢ ((A +o B) ∖ A) = {y∣∃x(x ∈ B ⋀ y = (A +o x))}
59	1, 3, 58	3eqtr4 1508	1 ⊢ ran F = ((A +o B) ∖ A)

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 146   ⋀ wa 223   = wceq 958   ∈ wcel 960  ∃wex 982  {cab 1466  ∃wrex 1649   ∖ cdif 2047   ⊆ wss 2050  {copab 2671  Ord word 2953  ωcom 3137  ran crn 3177  (class class class)co 3969   +_o coa 4136

This theorem is referenced by:  unfilem2 4561

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-oadd 4141

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