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 5669	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = (∅ × 𝐴))
2		0xp 5754	. . . 4 ⊢ (∅ × 𝐴) = ∅
3	1, 2	eqtrdi 2811	. . 3 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = ∅)
4		unieq 4878	. . . . 5 ⊢ ((𝐴 × 𝐴) = ∅ → ∪ (𝐴 × 𝐴) = ∪ ∅)
5	4	unieqd 4880	. . . 4 ⊢ ((𝐴 × 𝐴) = ∅ → ∪ ∪ (𝐴 × 𝐴) = ∪ ∪ ∅)
6		uni0 4896	. . . . . 6 ⊢ ∪ ∅ = ∅
7	6	unieqi 4879	. . . . 5 ⊢ ∪ ∪ ∅ = ∪ ∅
8	7, 6	eqtri 2783	. . . 4 ⊢ ∪ ∪ ∅ = ∅
9		eqtr 2780	. . . . 5 ⊢ ((∪ ∪ (𝐴 × 𝐴) = ∪ ∪ ∅ ∧ ∪ ∪ ∅ = ∅) → ∪ ∪ (𝐴 × 𝐴) = ∅)
10		eqtr 2780	. . . . . . 7 ⊢ ((∪ ∪ (𝐴 × 𝐴) = ∅ ∧ ∅ = 𝐴) → ∪ ∪ (𝐴 × 𝐴) = 𝐴)
11	10	expcom 419	. . . . . 6 ⊢ (∅ = 𝐴 → (∪ ∪ (𝐴 × 𝐴) = ∅ → ∪ ∪ (𝐴 × 𝐴) = 𝐴))
12	11	eqcoms 2768	. . . . 5 ⊢ (𝐴 = ∅ → (∪ ∪ (𝐴 × 𝐴) = ∅ → ∪ ∪ (𝐴 × 𝐴) = 𝐴))
13	9, 12	syl5com 32	. . . 4 ⊢ ((∪ ∪ (𝐴 × 𝐴) = ∪ ∪ ∅ ∧ ∪ ∪ ∅ = ∅) → (𝐴 = ∅ → ∪ ∪ (𝐴 × 𝐴) = 𝐴))
14	5, 8, 13	sylancl 598	. . 3 ⊢ ((𝐴 × 𝐴) = ∅ → (𝐴 = ∅ → ∪ ∪ (𝐴 × 𝐴) = 𝐴))
15	3, 14	mpcom 39	. 2 ⊢ (𝐴 = ∅ → ∪ ∪ (𝐴 × 𝐴) = 𝐴)
16		df-ne 2956	. . 3 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
17		xpnz 6151	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≠ ∅) ↔ (𝐴 × 𝐴) ≠ ∅)
18		unixp 6280	. . . . 5 ⊢ ((𝐴 × 𝐴) ≠ ∅ → ∪ ∪ (𝐴 × 𝐴) = (𝐴 ∪ 𝐴))
19		unidm 4104	. . . . 5 ⊢ (𝐴 ∪ 𝐴) = 𝐴
20	18, 19	eqtrdi 2811	. . . 4 ⊢ ((𝐴 × 𝐴) ≠ ∅ → ∪ ∪ (𝐴 × 𝐴) = 𝐴)
21	17, 20	sylbi 220	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≠ ∅) → ∪ ∪ (𝐴 × 𝐴) = 𝐴)
22	16, 16, 21	sylancbr 613	. 2 ⊢ (¬ 𝐴 = ∅ → ∪ ∪ (𝐴 × 𝐴) = 𝐴)
23	15, 22	pm2.61i 184	1 ⊢ ∪ ∪ (𝐴 × 𝐴) = 𝐴

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ≠ wne 2955   ∪ cun 3897  ∅c0 4279  ∪ cuni 4867   × cxp 5653

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-xp 5661  df-rel 5662  df-cnv 5663  df-dm 5665  df-rn 5666

This theorem is used by:  psss  18671