Theorem unixpid 6282

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 5686	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = (∅ × 𝐴))
2		0xp 5770	. . . 4 ⊢ (∅ × 𝐴) = ∅
3	1, 2	eqtrdi 2783	. . 3 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = ∅)
4		unieq 4914	. . . . 5 ⊢ ((𝐴 × 𝐴) = ∅ → ∪ (𝐴 × 𝐴) = ∪ ∅)
5	4	unieqd 4916	. . . 4 ⊢ ((𝐴 × 𝐴) = ∅ → ∪ ∪ (𝐴 × 𝐴) = ∪ ∪ ∅)
6		uni0 4933	. . . . . 6 ⊢ ∪ ∅ = ∅
7	6	unieqi 4915	. . . . 5 ⊢ ∪ ∪ ∅ = ∪ ∅
8	7, 6	eqtri 2755	. . . 4 ⊢ ∪ ∪ ∅ = ∅
9		eqtr 2750	. . . . 5 ⊢ ((∪ ∪ (𝐴 × 𝐴) = ∪ ∪ ∅ ∧ ∪ ∪ ∅ = ∅) → ∪ ∪ (𝐴 × 𝐴) = ∅)
10		eqtr 2750	. . . . . . 7 ⊢ ((∪ ∪ (𝐴 × 𝐴) = ∅ ∧ ∅ = 𝐴) → ∪ ∪ (𝐴 × 𝐴) = 𝐴)
11	10	expcom 413	. . . . . 6 ⊢ (∅ = 𝐴 → (∪ ∪ (𝐴 × 𝐴) = ∅ → ∪ ∪ (𝐴 × 𝐴) = 𝐴))
12	11	eqcoms 2735	. . . . 5 ⊢ (𝐴 = ∅ → (∪ ∪ (𝐴 × 𝐴) = ∅ → ∪ ∪ (𝐴 × 𝐴) = 𝐴))
13	9, 12	syl5com 31	. . . 4 ⊢ ((∪ ∪ (𝐴 × 𝐴) = ∪ ∪ ∅ ∧ ∪ ∪ ∅ = ∅) → (𝐴 = ∅ → ∪ ∪ (𝐴 × 𝐴) = 𝐴))
14	5, 8, 13	sylancl 585	. . 3 ⊢ ((𝐴 × 𝐴) = ∅ → (𝐴 = ∅ → ∪ ∪ (𝐴 × 𝐴) = 𝐴))
15	3, 14	mpcom 38	. 2 ⊢ (𝐴 = ∅ → ∪ ∪ (𝐴 × 𝐴) = 𝐴)
16		df-ne 2936	. . 3 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
17		xpnz 6157	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≠ ∅) ↔ (𝐴 × 𝐴) ≠ ∅)
18		unixp 6280	. . . . 5 ⊢ ((𝐴 × 𝐴) ≠ ∅ → ∪ ∪ (𝐴 × 𝐴) = (𝐴 ∪ 𝐴))
19		unidm 4148	. . . . 5 ⊢ (𝐴 ∪ 𝐴) = 𝐴
20	18, 19	eqtrdi 2783	. . . 4 ⊢ ((𝐴 × 𝐴) ≠ ∅ → ∪ ∪ (𝐴 × 𝐴) = 𝐴)
21	17, 20	sylbi 216	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≠ ∅) → ∪ ∪ (𝐴 × 𝐴) = 𝐴)
22	16, 16, 21	sylancbr 600	. 2 ⊢ (¬ 𝐴 = ∅ → ∪ ∪ (𝐴 × 𝐴) = 𝐴)
23	15, 22	pm2.61i 182	1 ⊢ ∪ ∪ (𝐴 × 𝐴) = 𝐴

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1534   ≠ wne 2935   ∪ cun 3942  ∅c0 4318  ∪ cuni 4903   × cxp 5670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-xp 5678  df-rel 5679  df-cnv 5680  df-dm 5682  df-rn 5683

This theorem is referenced by:  psss  18563