Theorem fneref 37353

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 2763	. . 3 ⊢ ∪ 𝐴 = ∪ 𝐴
2		ssid 3935	. . . . 5 ⊢ 𝑥 ⊆ 𝑥
3		elequ2 2162	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑥))
4		sseq1 3938	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑧 ⊆ 𝑥 ↔ 𝑥 ⊆ 𝑥))
5	3, 4	anbi12d 644	. . . . . 6 ⊢ (𝑧 = 𝑥 → ((𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥) ↔ (𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑥)))
6	5	rspcev 3580	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑥)) → ∃𝑧 ∈ 𝐴 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))
7	2, 6	mpanr2 717	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥) → ∃𝑧 ∈ 𝐴 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))
8	7	rgen2 3206	. . 3 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐴 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥)
9	1, 8	pm3.2i 476	. 2 ⊢ (∪ 𝐴 = ∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐴 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))
10	1, 1	isfne2 37345	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴Fne𝐴 ↔ (∪ 𝐴 = ∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐴 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))))
11	9, 10	mpbiri 261	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴Fne𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900  ∪ cuni 4872   class class class wbr 5108  Fnecfne 37339

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-iota 6520  df-fun 6568  df-fv 6574  df-topgen 17664  df-fne 37340

This theorem is used by: (None)