Theorem fness 36367

Description: A cover is finer than its subcovers. (Contributed by Jeff Hankins, 11-Oct-2009.)

Hypotheses

Ref	Expression
fness.1	⊢ 𝑋 = ∪ 𝐴
fness.2	⊢ 𝑌 = ∪ 𝐵

Assertion

Ref	Expression
fness	⊢ ((𝐵 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐵 ∧ 𝑋 = 𝑌) → 𝐴Fne𝐵)

Proof of Theorem fness

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

Step	Hyp	Ref	Expression
1		simp3 1138	. 2 ⊢ ((𝐵 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐵 ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌)
2		ssel2 3953	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵)
3	2	3adant3 1132	. . . . . 6 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥) → 𝑥 ∈ 𝐵)
4		simp3 1138	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑥)
5		ssid 3981	. . . . . . 7 ⊢ 𝑥 ⊆ 𝑥
6	4, 5	jctir 520	. . . . . 6 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥) → (𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑥))
7		elequ2 2123	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑥))
8		sseq1 3984	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝑧 ⊆ 𝑥 ↔ 𝑥 ⊆ 𝑥))
9	7, 8	anbi12d 632	. . . . . . 7 ⊢ (𝑧 = 𝑥 → ((𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥) ↔ (𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑥)))
10	9	rspcev 3601	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ⊆ 𝑥)) → ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))
11	3, 6, 10	syl2anc 584	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥) → ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))
12	11	3expib 1122	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥) → ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥)))
13	12	ralrimivv 3185	. . 3 ⊢ (𝐴 ⊆ 𝐵 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))
14	13	3ad2ant2 1134	. 2 ⊢ ((𝐵 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐵 ∧ 𝑋 = 𝑌) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))
15		fness.1	. . . 4 ⊢ 𝑋 = ∪ 𝐴
16		fness.2	. . . 4 ⊢ 𝑌 = ∪ 𝐵
17	15, 16	isfne2 36360	. . 3 ⊢ (𝐵 ∈ 𝐶 → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))))
18	17	3ad2ant1 1133	. 2 ⊢ ((𝐵 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐵 ∧ 𝑋 = 𝑌) → (𝐴Fne𝐵 ↔ (𝑋 = 𝑌 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝐵 (𝑦 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑥))))
19	1, 14, 18	mpbir2and 713	1 ⊢ ((𝐵 ∈ 𝐶 ∧ 𝐴 ⊆ 𝐵 ∧ 𝑋 = 𝑌) → 𝐴Fne𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  ∀wral 3051  ∃wrex 3060   ⊆ wss 3926  ∪ cuni 4883   class class class wbr 5119  Fnecfne 36354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-iota 6484  df-fun 6533  df-fv 6539  df-topgen 17457  df-fne 36355

This theorem is referenced by:  fnessref  36375  refssfne  36376