Theorem disjpss 1738

Description: A class is a proper subset of its union with a disjoint nonempty class.

Assertion

Ref	Expression
disjpss	⊢ (((A ∩ B) = ∅ ∧ ¬ B = ∅) → A ⊂ (A ∪ B))

Proof of Theorem disjpss

Step	Hyp	Ref	Expression
1		sseq2 1522	. . . . . 6 ⊢ ((A ∩ B) = ∅ → (B ⊆ (A ∩ B) ↔ B ⊆ ∅))
2		ssid 1519	. . . . . . . 8 ⊢ B ⊆ B
3	2	biantru 543	. . . . . . 7 ⊢ (B ⊆ A ↔ (B ⊆ A ∧ B ⊆ B))
4		ssin 1659	. . . . . . 7 ⊢ ((B ⊆ A ∧ B ⊆ B) ↔ B ⊆ (A ∩ B))
5	3, 4	bitr 151	. . . . . 6 ⊢ (B ⊆ A ↔ B ⊆ (A ∩ B))
6	1, 5	syl5bb 410	. . . . 5 ⊢ ((A ∩ B) = ∅ → (B ⊆ A ↔ B ⊆ ∅))
7		ss0 1727	. . . . 5 ⊢ (B ⊆ ∅ → B = ∅)
8	6, 7	syl6bi 187	. . . 4 ⊢ ((A ∩ B) = ∅ → (B ⊆ A → B = ∅))
9	8	con3d 87	. . 3 ⊢ ((A ∩ B) = ∅ → (¬ B = ∅ → ¬ B ⊆ A))
10	9	imp 277	. 2 ⊢ (((A ∩ B) = ∅ ∧ ¬ B = ∅) → ¬ B ⊆ A)
11		nsspssun 1666	. . 3 ⊢ (¬ B ⊆ A ↔ A ⊂ (B ∪ A))
12		uncom 1604	. . . 4 ⊢ (B ∪ A) = (A ∪ B)
13	12	psseq2i 1562	. . 3 ⊢ (A ⊂ (B ∪ A) ↔ A ⊂ (A ∪ B))
14	11, 13	bitr 151	. 2 ⊢ (¬ B ⊆ A ↔ A ⊂ (A ∪ B))
15	10, 14	sylib 173	1 ⊢ (((A ∩ B) = ∅ ∧ ¬ B = ∅) → A ⊂ (A ∪ B))

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196   = wceq 1091   ∪ cun 1485   ∩ cin 1486   ⊆ wss 1487   ⊂ wpss 1488  ∅c0 1707

This theorem is referenced by:  infxpidmlem11 4943

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708

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