Theorem gt1 474

Description: Part of Lemma 1 from Gaisi Takeuti, "Quantum Set Theory".

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 172	. . . . 5 b C d
4	3	comcom 435	. . . 4 d C b
5		gt1.3	. . . . . . 7 c ≤ d⊥
6	5	lecom 172	. . . . . 6 c C d⊥
7	6	comcom7 442	. . . . 5 c C d
8	7	comcom 435	. . . 4 d C c
9	4, 8	com2or 465	. . 3 d C (b ∪ c)
10	9	comcom 435	. 2 (b ∪ c) C d
11	1, 10	bctr 173	1 a C d

Syntax hints:   = wb 1   ≤ wle 2   C wc 3  ^⊥ wn 4   ∪ wo 6

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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