Theorem gt1 492

Description: Part of Lemma 1 from Gaisi Takeuti, "Quantum Set Theory". (Contributed by NM, 2-Dec-1998.)

Hypotheses

Ref	Expression
gt1.1	a = (b ∪ c)
gt1.2	b ≤ d
gt1.3	c ≤ d⊥

Assertion

Ref	Expression
gt1	a C d

Proof of Theorem gt1

Step	Hyp	Ref	Expression
1		gt1.1	. 2 a = (b ∪ c)
2		gt1.2	. . . . . 6 b ≤ d
3	2	lecom 180	. . . . 5 b C d
4	3	comcom 453	. . . 4 d C b
5		gt1.3	. . . . . . 7 c ≤ d⊥
6	5	lecom 180	. . . . . 6 c C d⊥
7	6	comcom7 460	. . . . 5 c C d
8	7	comcom 453	. . . 4 d C c
9	4, 8	com2or 483	. . 3 d C (b ∪ c)
10	9	comcom 453	. 2 (b ∪ c) C d
11	1, 10	bctr 181	1 a C d

This proof depends on syntax axioms:   = wb 1   ≤ wle 2   C wc 3  ^⊥ wn 4   ∪ wo 6

This proof depends on axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439

This proof depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is used by: (None)