Theorem unidif 3945

Description: If the difference 𝐴 ∖ 𝐵 contains the largest members of 𝐴, then the union of the difference is the union of 𝐴. (Contributed by NM, 22-Mar-2004.)

Assertion

Ref	Expression
unidif	⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝑥 ⊆ 𝑦 → ∪ (𝐴 ∖ 𝐵) = ∪ 𝐴)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem unidif

Step	Hyp	Ref	Expression
1		uniss2 3944	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝑥 ⊆ 𝑦 → ∪ 𝐴 ⊆ ∪ (𝐴 ∖ 𝐵))
2		difss 3344	. . . 4 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
3	2	unissi 3936	. . 3 ⊢ ∪ (𝐴 ∖ 𝐵) ⊆ ∪ 𝐴
4	1, 3	jctil 312	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝑥 ⊆ 𝑦 → (∪ (𝐴 ∖ 𝐵) ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ ∪ (𝐴 ∖ 𝐵)))
5		eqss 3252	. 2 ⊢ (∪ (𝐴 ∖ 𝐵) = ∪ 𝐴 ↔ (∪ (𝐴 ∖ 𝐵) ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ ∪ (𝐴 ∖ 𝐵)))
6	4, 5	sylibr 134	1 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝑥 ⊆ 𝑦 → ∪ (𝐴 ∖ 𝐵) = ∪ 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398  ∀wral 2520  ∃wrex 2521   ∖ cdif 3207   ⊆ wss 3210  ∪ cuni 3913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-dif 3212  df-in 3216  df-ss 3223  df-uni 3914

This theorem is referenced by: (None)