Theorem unixpid 6129

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 5563	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = (∅ × 𝐴))
2		0xp 5643	. . . 4 ⊢ (∅ × 𝐴) = ∅
3	1, 2	syl6eq 2872	. . 3 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = ∅)
4		unieq 4839	. . . . 5 ⊢ ((𝐴 × 𝐴) = ∅ → ∪ (𝐴 × 𝐴) = ∪ ∅)
5	4	unieqd 4841	. . . 4 ⊢ ((𝐴 × 𝐴) = ∅ → ∪ ∪ (𝐴 × 𝐴) = ∪ ∪ ∅)
6		uni0 4858	. . . . . 6 ⊢ ∪ ∅ = ∅
7	6	unieqi 4840	. . . . 5 ⊢ ∪ ∪ ∅ = ∪ ∅
8	7, 6	eqtri 2844	. . . 4 ⊢ ∪ ∪ ∅ = ∅
9		eqtr 2841	. . . . 5 ⊢ ((∪ ∪ (𝐴 × 𝐴) = ∪ ∪ ∅ ∧ ∪ ∪ ∅ = ∅) → ∪ ∪ (𝐴 × 𝐴) = ∅)
10		eqtr 2841	. . . . . . 7 ⊢ ((∪ ∪ (𝐴 × 𝐴) = ∅ ∧ ∅ = 𝐴) → ∪ ∪ (𝐴 × 𝐴) = 𝐴)
11	10	expcom 416	. . . . . 6 ⊢ (∅ = 𝐴 → (∪ ∪ (𝐴 × 𝐴) = ∅ → ∪ ∪ (𝐴 × 𝐴) = 𝐴))
12	11	eqcoms 2829	. . . . 5 ⊢ (𝐴 = ∅ → (∪ ∪ (𝐴 × 𝐴) = ∅ → ∪ ∪ (𝐴 × 𝐴) = 𝐴))
13	9, 12	syl5com 31	. . . 4 ⊢ ((∪ ∪ (𝐴 × 𝐴) = ∪ ∪ ∅ ∧ ∪ ∪ ∅ = ∅) → (𝐴 = ∅ → ∪ ∪ (𝐴 × 𝐴) = 𝐴))
14	5, 8, 13	sylancl 588	. . 3 ⊢ ((𝐴 × 𝐴) = ∅ → (𝐴 = ∅ → ∪ ∪ (𝐴 × 𝐴) = 𝐴))
15	3, 14	mpcom 38	. 2 ⊢ (𝐴 = ∅ → ∪ ∪ (𝐴 × 𝐴) = 𝐴)
16		df-ne 3017	. . 3 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
17		xpnz 6010	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≠ ∅) ↔ (𝐴 × 𝐴) ≠ ∅)
18		unixp 6127	. . . . 5 ⊢ ((𝐴 × 𝐴) ≠ ∅ → ∪ ∪ (𝐴 × 𝐴) = (𝐴 ∪ 𝐴))
19		unidm 4127	. . . . 5 ⊢ (𝐴 ∪ 𝐴) = 𝐴
20	18, 19	syl6eq 2872	. . . 4 ⊢ ((𝐴 × 𝐴) ≠ ∅ → ∪ ∪ (𝐴 × 𝐴) = 𝐴)
21	17, 20	sylbi 219	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ≠ ∅) → ∪ ∪ (𝐴 × 𝐴) = 𝐴)
22	16, 16, 21	sylancbr 602	. 2 ⊢ (¬ 𝐴 = ∅ → ∪ ∪ (𝐴 × 𝐴) = 𝐴)
23	15, 22	pm2.61i 184	1 ⊢ ∪ ∪ (𝐴 × 𝐴) = 𝐴

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1533   ≠ wne 3016   ∪ cun 3933  ∅c0 4290  ∪ cuni 4831   × cxp 5547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-xp 5555  df-rel 5556  df-cnv 5557  df-dm 5559  df-rn 5560

This theorem is referenced by:  psss  17818