Theorem unixpid 5670

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

Assertion

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

Proof of Theorem unixpid

Step	Hyp	Ref	Expression
1		xpeq1 5128	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = (∅ × 𝐴))
2		0xp 5199	. . . 4 ⊢ (∅ × 𝐴) = ∅
3	1, 2	syl6eq 2672	. . 3 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = ∅)
4		unieq 4444	. . . . 5 ⊢ ((𝐴 × 𝐴) = ∅ → ∪ (𝐴 × 𝐴) = ∪ ∅)
5	4	unieqd 4446	. . . 4 ⊢ ((𝐴 × 𝐴) = ∅ → ∪ ∪ (𝐴 × 𝐴) = ∪ ∪ ∅)
6		uni0 4465	. . . . . 6 ⊢ ∪ ∅ = ∅
7	6	unieqi 4445	. . . . 5 ⊢ ∪ ∪ ∅ = ∪ ∅
8	7, 6	eqtri 2644	. . . 4 ⊢ ∪ ∪ ∅ = ∅
9		eqtr 2641	. . . . 5 ⊢ ((∪ ∪ (𝐴 × 𝐴) = ∪ ∪ ∅ ∧ ∪ ∪ ∅ = ∅) → ∪ ∪ (𝐴 × 𝐴) = ∅)
10		eqtr 2641	. . . . . . 7 ⊢ ((∪ ∪ (𝐴 × 𝐴) = ∅ ∧ ∅ = 𝐴) → ∪ ∪ (𝐴 × 𝐴) = 𝐴)
11	10	expcom 451	. . . . . 6 ⊢ (∅ = 𝐴 → (∪ ∪ (𝐴 × 𝐴) = ∅ → ∪ ∪ (𝐴 × 𝐴) = 𝐴))
12	11	eqcoms 2630	. . . . 5 ⊢ (𝐴 = ∅ → (∪ ∪ (𝐴 × 𝐴) = ∅ → ∪ ∪ (𝐴 × 𝐴) = 𝐴))
13	9, 12	syl5com 31	. . . 4 ⊢ ((∪ ∪ (𝐴 × 𝐴) = ∪ ∪ ∅ ∧ ∪ ∪ ∅ = ∅) → (𝐴 = ∅ → ∪ ∪ (𝐴 × 𝐴) = 𝐴))
14	5, 8, 13	sylancl 694	. . 3 ⊢ ((𝐴 × 𝐴) = ∅ → (𝐴 = ∅ → ∪ ∪ (𝐴 × 𝐴) = 𝐴))
15	3, 14	mpcom 38	. 2 ⊢ (𝐴 = ∅ → ∪ ∪ (𝐴 × 𝐴) = 𝐴)
16		df-ne 2795	. . 3 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
17		xpnz 5553	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≠ ∅) ↔ (𝐴 × 𝐴) ≠ ∅)
18		unixp 5668	. . . . 5 ⊢ ((𝐴 × 𝐴) ≠ ∅ → ∪ ∪ (𝐴 × 𝐴) = (𝐴 ∪ 𝐴))
19		unidm 3756	. . . . 5 ⊢ (𝐴 ∪ 𝐴) = 𝐴
20	18, 19	syl6eq 2672	. . . 4 ⊢ ((𝐴 × 𝐴) ≠ ∅ → ∪ ∪ (𝐴 × 𝐴) = 𝐴)
21	17, 20	sylbi 207	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≠ ∅) → ∪ ∪ (𝐴 × 𝐴) = 𝐴)
22	16, 16, 21	sylancbr 700	. 2 ⊢ (¬ 𝐴 = ∅ → ∪ ∪ (𝐴 × 𝐴) = 𝐴)
23	15, 22	pm2.61i 176	1 ⊢ ∪ ∪ (𝐴 × 𝐴) = 𝐴

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1483   ≠ wne 2794   ∪ cun 3572  ∅c0 3915  ∪ cuni 4436   × cxp 5112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125

This theorem is referenced by:  psss  17214