Theorem fneref 31515

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 2610	. . 3 ⊢ ∪ 𝐴 = ∪ 𝐴
2		ssid 3587	. . . . 5 ⊢ 𝑥 ⊆ 𝑥
3		elequ2 1991	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑥))
4		sseq1 3589	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑧 ⊆ 𝑥 ↔ 𝑥 ⊆ 𝑥))
5	3, 4	anbi12d 743	. . . . . 6 ⊢ (𝑧 = 𝑥 → ((𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥) ↔ (𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑥)))
6	5	rspcev 3282	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑥)) → ∃𝑧 ∈ 𝐴 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))
7	2, 6	mpanr2 716	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥) → ∃𝑧 ∈ 𝐴 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))
8	7	rgen2 2958	. . 3 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐴 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥)
9	1, 8	pm3.2i 470	. 2 ⊢ (∪ 𝐴 = ∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐴 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))
10	1, 1	isfne2 31507	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴Fne𝐴 ↔ (∪ 𝐴 = ∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐴 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))))
11	9, 10	mpbiri 247	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴Fne𝐴)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540  ∪ cuni 4372   class class class wbr 4583  Fnecfne 31501

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-topgen 15927  df-fne 31502

This theorem is referenced by: (None)