Theorem xpsnen 3339

Description: A set is equinumerous to its cross-product with a singleton. Proposition 4.22(c) of [Mendelson] p. 254.

Hypotheses

Ref	Expression
xpsnen.1	⊢ A ∈ V
xpsnen.2	⊢ B ∈ V

Assertion

Ref	Expression
xpsnen	⊢ (A × {B}) ≈ A

Proof of Theorem xpsnen

Step	Hyp	Ref	Expression
1		xpsnen.1	. . 3 ⊢ A ∈ V
2		snex 1859	. . 3 ⊢ {B} ∈ V
3	1, 2	xpex 2488	. 2 ⊢ (A × {B}) ∈ V
4		elxp 2442	. . 3 ⊢ (y ∈ (A × {B}) ↔ ∃x∃z(y = ⟨x, z⟩ ∧ (x ∈ A ∧ z ∈ {B})))
5		visset 1350	. . . . . 6 ⊢ x ∈ V
6		inteq 1968	. . . . . . . . 9 ⊢ (y = ⟨x, z⟩ → ∩y = ∩⟨x, z⟩)
7	6	inteqd 1970	. . . . . . . 8 ⊢ (y = ⟨x, z⟩ → ∩∩y = ∩∩⟨x, z⟩)
8	5	op1stb 1992	. . . . . . . 8 ⊢ ∩∩⟨x, z⟩ = x
9	7, 8	syl6eq 1140	. . . . . . 7 ⊢ (y = ⟨x, z⟩ → ∩∩y = x)
10	9	eleq1d 1155	. . . . . 6 ⊢ (y = ⟨x, z⟩ → (∩∩y ∈ V ↔ x ∈ V))
11	5, 10	mpbiri 169	. . . . 5 ⊢ (y = ⟨x, z⟩ → ∩∩y ∈ V)
12	11	adantr 306	. . . 4 ⊢ ((y = ⟨x, z⟩ ∧ (x ∈ A ∧ z ∈ {B})) → ∩∩y ∈ V)
13	12	19.23aivv 953	. . 3 ⊢ (∃x∃z(y = ⟨x, z⟩ ∧ (x ∈ A ∧ z ∈ {B})) → ∩∩y ∈ V)
14	4, 13	sylbi 174	. 2 ⊢ (y ∈ (A × {B}) → ∩∩y ∈ V)
15		opex 1893	. . 3 ⊢ ⟨x, B⟩ ∈ V
16	15	a1i 7	. 2 ⊢ (x ∈ A → ⟨x, B⟩ ∈ V)
17		eleq1 1149	. . . . . 6 ⊢ (x = ∩∩y → (x ∈ V ↔ ∩∩y ∈ V))
18	5, 17	mpbii 168	. . . . 5 ⊢ (x = ∩∩y → ∩∩y ∈ V)
19		opeq1 1876	. . . . . . . . 9 ⊢ (x = ∩∩y → ⟨x, B⟩ = ⟨∩∩y, B⟩)
20	19	cleq2d 1112	. . . . . . . 8 ⊢ (x = ∩∩y → (y = ⟨x, B⟩ ↔ y = ⟨∩∩y, B⟩))
21		eleq1 1149	. . . . . . . 8 ⊢ (x = ∩∩y → (x ∈ A ↔ ∩∩y ∈ A))
22	20, 21	anbi12d 476	. . . . . . 7 ⊢ (x = ∩∩y → ((y = ⟨x, B⟩ ∧ x ∈ A) ↔ (y = ⟨∩∩y, B⟩ ∧ ∩∩y ∈ A)))
23	22	ceqsexgv 1412	. . . . . 6 ⊢ (∩∩y ∈ V → (∃x(x = ∩∩y ∧ (y = ⟨x, B⟩ ∧ x ∈ A)) ↔ (y = ⟨∩∩y, B⟩ ∧ ∩∩y ∈ A)))
24		ancom 333	. . . . . . . . . . 11 ⊢ (((y = ⟨x, z⟩ ∧ x ∈ A) ∧ z ∈ {B}) ↔ (z ∈ {B} ∧ (y = ⟨x, z⟩ ∧ x ∈ A)))
25		anass 336	. . . . . . . . . . 11 ⊢ (((y = ⟨x, z⟩ ∧ x ∈ A) ∧ z ∈ {B}) ↔ (y = ⟨x, z⟩ ∧ (x ∈ A ∧ z ∈ {B})))
26		elsn 1820	. . . . . . . . . . . 12 ⊢ (z ∈ {B} ↔ z = B)
27	26	anbi1i 368	. . . . . . . . . . 11 ⊢ ((z ∈ {B} ∧ (y = ⟨x, z⟩ ∧ x ∈ A)) ↔ (z = B ∧ (y = ⟨x, z⟩ ∧ x ∈ A)))
28	24, 25, 27	3bitr3 156	. . . . . . . . . 10 ⊢ ((y = ⟨x, z⟩ ∧ (x ∈ A ∧ z ∈ {B})) ↔ (z = B ∧ (y = ⟨x, z⟩ ∧ x ∈ A)))
29	28	biex 733	. . . . . . . . 9 ⊢ (∃z(y = ⟨x, z⟩ ∧ (x ∈ A ∧ z ∈ {B})) ↔ ∃z(z = B ∧ (y = ⟨x, z⟩ ∧ x ∈ A)))
30		xpsnen.2	. . . . . . . . . 10 ⊢ B ∈ V
31		opeq2 1877	. . . . . . . . . . . 12 ⊢ (z = B → ⟨x, z⟩ = ⟨x, B⟩)
32	31	cleq2d 1112	. . . . . . . . . . 11 ⊢ (z = B → (y = ⟨x, z⟩ ↔ y = ⟨x, B⟩))
33	32	anbi1d 469	. . . . . . . . . 10 ⊢ (z = B → ((y = ⟨x, z⟩ ∧ x ∈ A) ↔ (y = ⟨x, B⟩ ∧ x ∈ A)))
34	30, 33	ceqsexv 1371	. . . . . . . . 9 ⊢ (∃z(z = B ∧ (y = ⟨x, z⟩ ∧ x ∈ A)) ↔ (y = ⟨x, B⟩ ∧ x ∈ A))
35		inteq 1968	. . . . . . . . . . . . . 14 ⊢ (y = ⟨x, B⟩ → ∩y = ∩⟨x, B⟩)
36	35	inteqd 1970	. . . . . . . . . . . . 13 ⊢ (y = ⟨x, B⟩ → ∩∩y = ∩∩⟨x, B⟩)
37	5	op1stb 1992	. . . . . . . . . . . . 13 ⊢ ∩∩⟨x, B⟩ = x
38	36, 37	syl6req 1141	. . . . . . . . . . . 12 ⊢ (y = ⟨x, B⟩ → x = ∩∩y)
39	38	pm4.71ri 484	. . . . . . . . . . 11 ⊢ (y = ⟨x, B⟩ ↔ (x = ∩∩y ∧ y = ⟨x, B⟩))
40	39	anbi1i 368	. . . . . . . . . 10 ⊢ ((y = ⟨x, B⟩ ∧ x ∈ A) ↔ ((x = ∩∩y ∧ y = ⟨x, B⟩) ∧ x ∈ A))
41		anass 336	. . . . . . . . . 10 ⊢ (((x = ∩∩y ∧ y = ⟨x, B⟩) ∧ x ∈ A) ↔ (x = ∩∩y ∧ (y = ⟨x, B⟩ ∧ x ∈ A)))
42	40, 41	bitr 151	. . . . . . . . 9 ⊢ ((y = ⟨x, B⟩ ∧ x ∈ A) ↔ (x = ∩∩y ∧ (y = ⟨x, B⟩ ∧ x ∈ A)))
43	29, 34, 42	3bitr 155	. . . . . . . 8 ⊢ (∃z(y = ⟨x, z⟩ ∧ (x ∈ A ∧ z ∈ {B})) ↔ (x = ∩∩y ∧ (y = ⟨x, B⟩ ∧ x ∈ A)))
44	43	biex 733	. . . . . . 7 ⊢ (∃x∃z(y = ⟨x, z⟩ ∧ (x ∈ A ∧ z ∈ {B})) ↔ ∃x(x = ∩∩y ∧ (y = ⟨x, B⟩ ∧ x ∈ A)))
45	4, 44	bitr 151	. . . . . 6 ⊢ (y ∈ (A × {B}) ↔ ∃x(x = ∩∩y ∧ (y = ⟨x, B⟩ ∧ x ∈ A)))
46	23, 45	syl5bb 410	. . . . 5 ⊢ (∩∩y ∈ V → (y ∈ (A × {B}) ↔ (y = ⟨∩∩y, B⟩ ∧ ∩∩y ∈ A)))
47	18, 46	syl 12	. . . 4 ⊢ (x = ∩∩y → (y ∈ (A × {B}) ↔ (y = ⟨∩∩y, B⟩ ∧ ∩∩y ∈ A)))
48	47	pm5.32ri 490	. . 3 ⊢ ((y ∈ (A × {B}) ∧ x = ∩∩y) ↔ ((y = ⟨∩∩y, B⟩ ∧ ∩∩y ∈ A) ∧ x = ∩∩y))
49	38	adantr 306	. . . . 5 ⊢ ((y = ⟨x, B⟩ ∧ x ∈ A) → x = ∩∩y)
50	49	pm4.71i 483	. . . 4 ⊢ ((y = ⟨x, B⟩ ∧ x ∈ A) ↔ ((y = ⟨x, B⟩ ∧ x ∈ A) ∧ x = ∩∩y))
51	22	pm5.32ri 490	. . . 4 ⊢ (((y = ⟨x, B⟩ ∧ x ∈ A) ∧ x = ∩∩y) ↔ ((y = ⟨∩∩y, B⟩ ∧ ∩∩y ∈ A) ∧ x = ∩∩y))
52	50, 51	bitr2 152	. . 3 ⊢ (((y = ⟨∩∩y, B⟩ ∧ ∩∩y ∈ A) ∧ x = ∩∩y) ↔ (y = ⟨x, B⟩ ∧ x ∈ A))
53		ancom 333	. . 3 ⊢ ((y = ⟨x, B⟩ ∧ x ∈ A) ↔ (x ∈ A ∧ y = ⟨x, B⟩))
54	48, 52, 53	3bitr 155	. 2 ⊢ ((y ∈ (A × {B}) ∧ x = ∩∩y) ↔ (x ∈ A ∧ y = ⟨x, B⟩))
55	3, 14, 16, 54	en2 3305	1 ⊢ (A × {B}) ≈ A

Syntax hints:   ↔ wb 127   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  Vcvv 1348  {csn 1808  ⟨cop 1810  ∩cint 1965   class class class wbr 2054   × cxp 2408   ≈ cen 3271

This theorem is referenced by:  xpsneng 3340  endisj 3341  xpdom3 3347  unxpdom2 3651  sucxpdom 3652  uncdadom 3718  cdaen 3719  cda0en 3720  cda1en 3721  xp1en 3722  cdacomen 3724  cdaassen 3725  cdadom1 3727  xpnnen 4927

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-int 1966  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-en 3274

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