Theorem fneref 36551

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 2737	. . 3 ⊢ ∪ 𝐴 = ∪ 𝐴
2		ssid 3945	. . . . 5 ⊢ 𝑥 ⊆ 𝑥
3		elequ2 2129	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑥))
4		sseq1 3948	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑧 ⊆ 𝑥 ↔ 𝑥 ⊆ 𝑥))
5	3, 4	anbi12d 633	. . . . . 6 ⊢ (𝑧 = 𝑥 → ((𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥) ↔ (𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑥)))
6	5	rspcev 3565	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑥)) → ∃𝑧 ∈ 𝐴 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))
7	2, 6	mpanr2 705	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥) → ∃𝑧 ∈ 𝐴 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))
8	7	rgen2 3178	. . 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 36543	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴Fne𝐴 ↔ (∪ 𝐴 = ∪ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐴 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))))
11	9, 10	mpbiri 258	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴Fne𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ⊆ wss 3890  ∪ cuni 4851   class class class wbr 5086  Fnecfne 36537

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6449  df-fun 6495  df-fv 6501  df-topgen 17400  df-fne 36538

This theorem is referenced by: (None)