Theorem fneref 35223

Description: Reflexivity of the fineness relation. (Contributed by Jeff Hankins, 12-Oct-2009.)

Assertion

Ref	Expression
fneref	⊢ (𝐴 ∈ 𝑉 → 𝐴Fne𝐴)

Proof of Theorem fneref

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2732	. . 3 ⊢ ∪ 𝐴 = ∪ 𝐴
2		ssid 4003	. . . . 5 ⊢ 𝑥 ⊆ 𝑥
3		elequ2 2121	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑥))
4		sseq1 4006	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑧 ⊆ 𝑥 ↔ 𝑥 ⊆ 𝑥))
5	3, 4	anbi12d 631	. . . . . 6 ⊢ (𝑧 = 𝑥 → ((𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥) ↔ (𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑥)))
6	5	rspcev 3612	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑥)) → ∃𝑧 ∈ 𝐴 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))
7	2, 6	mpanr2 702	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥) → ∃𝑧 ∈ 𝐴 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))
8	7	rgen2 3197	. . 3 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐴 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥)
9	1, 8	pm3.2i 471	. 2 ⊢ (∪ 𝐴 = ∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐴 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))
10	1, 1	isfne2 35215	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴Fne𝐴 ↔ (∪ 𝐴 = ∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐴 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))))
11	9, 10	mpbiri 257	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴Fne𝐴)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061  ∃wrex 3070   ⊆ wss 3947  ∪ cuni 4907   class class class wbr 5147  Fnecfne 35209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-topgen 17385  df-fne 35210

This theorem is referenced by: (None)