Theorem frirr 2176

Description: A founded relation is irreflexive. Special case of Proposition 6.23 of [TakeutiZaring] p. 30.

Assertion

Ref	Expression
frirr	⊢ ((R Fr A ∧ x ∈ A) → ¬ xRx)

Distinct variable group(s):   x,R

Proof of Theorem frirr

Step	Hyp	Ref	Expression
1		visset 1350	. . . . 5 ⊢ x ∈ V
2	1	snnz 1846	. . . 4 ⊢ ¬ {x} = ∅
3		snex 1859	. . . . . . 7 ⊢ {x} ∈ V
4	3	frc 2172	. . . . . 6 ⊢ (R Fr A → (({x} ⊆ A ∧ ¬ {x} = ∅) → ∃y ∈ {x} ({x} ∩ {z∣zRy}) = ∅))
5	4	exp3a 292	. . . . 5 ⊢ (R Fr A → ({x} ⊆ A → (¬ {x} = ∅ → ∃y ∈ {x} ({x} ∩ {z∣zRy}) = ∅)))
6	1	snss 1849	. . . . 5 ⊢ (x ∈ A ↔ {x} ⊆ A)
7	5, 6	syl5ib 181	. . . 4 ⊢ (R Fr A → (x ∈ A → (¬ {x} = ∅ → ∃y ∈ {x} ({x} ∩ {z∣zRy}) = ∅)))
8	2, 7	mpii 45	. . 3 ⊢ (R Fr A → (x ∈ A → ∃y ∈ {x} ({x} ∩ {z∣zRy}) = ∅))
9		elsn 1820	. . . . 5 ⊢ (y ∈ {x} ↔ y = x)
10		breq2 2066	. . . . . . . . 9 ⊢ (y = x → (zRy ↔ zRx))
11	10	biabdv 1183	. . . . . . . 8 ⊢ (y = x → {z∣zRy} = {z∣zRx})
12	11	ineq2d 1645	. . . . . . 7 ⊢ (y = x → ({x} ∩ {z∣zRy}) = ({x} ∩ {z∣zRx}))
13	12	cleq1d 1109	. . . . . 6 ⊢ (y = x → (({x} ∩ {z∣zRy}) = ∅ ↔ ({x} ∩ {z∣zRx}) = ∅))
14		breq1 2065	. . . . . . . . . . . 12 ⊢ (z = x → (zRx ↔ xRx))
15	1, 14	elab 1415	. . . . . . . . . . 11 ⊢ (x ∈ {z∣zRx} ↔ xRx)
16	15	biimpr 134	. . . . . . . . . 10 ⊢ (xRx → x ∈ {z∣zRx})
17	1	snid 1830	. . . . . . . . . 10 ⊢ x ∈ {x}
18	16, 17	jctil 240	. . . . . . . . 9 ⊢ (xRx → (x ∈ {x} ∧ x ∈ {z∣zRx}))
19		elin 1635	. . . . . . . . 9 ⊢ (x ∈ ({x} ∩ {z∣zRx}) ↔ (x ∈ {x} ∧ x ∈ {z∣zRx}))
20	18, 19	sylibr 175	. . . . . . . 8 ⊢ (xRx → x ∈ ({x} ∩ {z∣zRx}))
21		n0i 1712	. . . . . . . 8 ⊢ (x ∈ ({x} ∩ {z∣zRx}) → ¬ ({x} ∩ {z∣zRx}) = ∅)
22	20, 21	syl 12	. . . . . . 7 ⊢ (xRx → ¬ ({x} ∩ {z∣zRx}) = ∅)
23	22	con2i 89	. . . . . 6 ⊢ (({x} ∩ {z∣zRx}) = ∅ → ¬ xRx)
24	13, 23	syl6bi 187	. . . . 5 ⊢ (y = x → (({x} ∩ {z∣zRy}) = ∅ → ¬ xRx))
25	9, 24	sylbi 174	. . . 4 ⊢ (y ∈ {x} → (({x} ∩ {z∣zRy}) = ∅ → ¬ xRx))
26	25	r19.23aiv 1284	. . 3 ⊢ (∃y ∈ {x} ({x} ∩ {z∣zRy}) = ∅ → ¬ xRx)
27	8, 26	syl6 23	. 2 ⊢ (R Fr A → (x ∈ A → ¬ xRx))
28	27	imp 277	1 ⊢ ((R Fr A ∧ x ∈ A) → ¬ xRx)

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196   = weq 797  {cab 1090   = wceq 1091   ∈ wcel 1092  ∃wrex 1202   ∩ cin 1486   ⊆ wss 1487  ∅c0 1707  {csn 1808   class class class wbr 2054   Fr wfr 2061

This theorem is referenced by:  efrirr 2180  dfwe2 2187

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-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

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-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-fr 2169

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