Theorem unidif 3776

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 3775	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝑥 ⊆ 𝑦 → ∪ 𝐴 ⊆ ∪ (𝐴 ∖ 𝐵))
2		difss 3207	. . . 4 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
3	2	unissi 3767	. . 3 ⊢ ∪ (𝐴 ∖ 𝐵) ⊆ ∪ 𝐴
4	1, 3	jctil 310	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝑥 ⊆ 𝑦 → (∪ (𝐴 ∖ 𝐵) ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ ∪ (𝐴 ∖ 𝐵)))
5		eqss 3117	. 2 ⊢ (∪ (𝐴 ∖ 𝐵) = ∪ 𝐴 ↔ (∪ (𝐴 ∖ 𝐵) ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ ∪ (𝐴 ∖ 𝐵)))
6	4, 5	sylibr 133	1 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝑥 ⊆ 𝑦 → ∪ (𝐴 ∖ 𝐵) = ∪ 𝐴)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332  ∀wral 2417  ∃wrex 2418   ∖ cdif 3073   ⊆ wss 3076  ∪ cuni 3744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-dif 3078  df-in 3082  df-ss 3089  df-uni 3745

This theorem is referenced by: (None)