Theorem unixpid 6248

Description: Field of a Cartesian square. (Contributed by FL, 10-Oct-2009.)

Assertion

Ref	Expression
unixpid	⊢ ∪ ∪ (𝐴 × 𝐴) = 𝐴

Proof of Theorem unixpid

Step	Hyp	Ref	Expression
1		xpeq1 5645	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = (∅ × 𝐴))
2		0xp 5730	. . . 4 ⊢ (∅ × 𝐴) = ∅
3	1, 2	eqtrdi 2787	. . 3 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = ∅)
4		unieq 4861	. . . . 5 ⊢ ((𝐴 × 𝐴) = ∅ → ∪ (𝐴 × 𝐴) = ∪ ∅)
5	4	unieqd 4863	. . . 4 ⊢ ((𝐴 × 𝐴) = ∅ → ∪ ∪ (𝐴 × 𝐴) = ∪ ∪ ∅)
6		uni0 4878	. . . . . 6 ⊢ ∪ ∅ = ∅
7	6	unieqi 4862	. . . . 5 ⊢ ∪ ∪ ∅ = ∪ ∅
8	7, 6	eqtri 2759	. . . 4 ⊢ ∪ ∪ ∅ = ∅
9		eqtr 2756	. . . . 5 ⊢ ((∪ ∪ (𝐴 × 𝐴) = ∪ ∪ ∅ ∧ ∪ ∪ ∅ = ∅) → ∪ ∪ (𝐴 × 𝐴) = ∅)
10		eqtr 2756	. . . . . . 7 ⊢ ((∪ ∪ (𝐴 × 𝐴) = ∅ ∧ ∅ = 𝐴) → ∪ ∪ (𝐴 × 𝐴) = 𝐴)
11	10	expcom 413	. . . . . 6 ⊢ (∅ = 𝐴 → (∪ ∪ (𝐴 × 𝐴) = ∅ → ∪ ∪ (𝐴 × 𝐴) = 𝐴))
12	11	eqcoms 2744	. . . . 5 ⊢ (𝐴 = ∅ → (∪ ∪ (𝐴 × 𝐴) = ∅ → ∪ ∪ (𝐴 × 𝐴) = 𝐴))
13	9, 12	syl5com 31	. . . 4 ⊢ ((∪ ∪ (𝐴 × 𝐴) = ∪ ∪ ∅ ∧ ∪ ∪ ∅ = ∅) → (𝐴 = ∅ → ∪ ∪ (𝐴 × 𝐴) = 𝐴))
14	5, 8, 13	sylancl 587	. . 3 ⊢ ((𝐴 × 𝐴) = ∅ → (𝐴 = ∅ → ∪ ∪ (𝐴 × 𝐴) = 𝐴))
15	3, 14	mpcom 38	. 2 ⊢ (𝐴 = ∅ → ∪ ∪ (𝐴 × 𝐴) = 𝐴)
16		df-ne 2933	. . 3 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
17		xpnz 6123	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≠ ∅) ↔ (𝐴 × 𝐴) ≠ ∅)
18		unixp 6246	. . . . 5 ⊢ ((𝐴 × 𝐴) ≠ ∅ → ∪ ∪ (𝐴 × 𝐴) = (𝐴 ∪ 𝐴))
19		unidm 4097	. . . . 5 ⊢ (𝐴 ∪ 𝐴) = 𝐴
20	18, 19	eqtrdi 2787	. . . 4 ⊢ ((𝐴 × 𝐴) ≠ ∅ → ∪ ∪ (𝐴 × 𝐴) = 𝐴)
21	17, 20	sylbi 217	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≠ ∅) → ∪ ∪ (𝐴 × 𝐴) = 𝐴)
22	16, 16, 21	sylancbr 602	. 2 ⊢ (¬ 𝐴 = ∅ → ∪ ∪ (𝐴 × 𝐴) = 𝐴)
23	15, 22	pm2.61i 182	1 ⊢ ∪ ∪ (𝐴 × 𝐴) = 𝐴

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ≠ wne 2932   ∪ cun 3887  ∅c0 4273  ∪ cuni 4850   × cxp 5629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-xp 5637  df-rel 5638  df-cnv 5639  df-dm 5641  df-rn 5642

This theorem is referenced by:  psss  18546