Theorem cvbr2 31801

Description: Binary relation expressing 𝐵 covers 𝐴. Definition of covers in [Kalmbach] p. 15. (Contributed by NM, 9-Jun-2004.) (New usage is discouraged.)

Assertion

Ref	Expression
cvbr2	⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∧ ∀𝑥 ∈ Cℋ ((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝑥 = 𝐵))))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem cvbr2

Step	Hyp	Ref	Expression
1		cvbr 31800	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∧ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵))))
2		iman 400	. . . . . 6 ⊢ (((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝑥 = 𝐵) ↔ ¬ ((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ ¬ 𝑥 = 𝐵))
3		anass 467	. . . . . . 7 ⊢ (((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ ¬ 𝑥 = 𝐵) ↔ (𝐴 ⊊ 𝑥 ∧ (𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 = 𝐵)))
4		dfpss2 4086	. . . . . . . 8 ⊢ (𝑥 ⊊ 𝐵 ↔ (𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 = 𝐵))
5	4	anbi2i 621	. . . . . . 7 ⊢ ((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵) ↔ (𝐴 ⊊ 𝑥 ∧ (𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 = 𝐵)))
6	3, 5	bitr4i 277	. . . . . 6 ⊢ (((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ ¬ 𝑥 = 𝐵) ↔ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵))
7	2, 6	xchbinx 333	. . . . 5 ⊢ (((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝑥 = 𝐵) ↔ ¬ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵))
8	7	ralbii 3091	. . . 4 ⊢ (∀𝑥 ∈ Cℋ ((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝑥 = 𝐵) ↔ ∀𝑥 ∈ Cℋ ¬ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵))
9		ralnex 3070	. . . 4 ⊢ (∀𝑥 ∈ Cℋ ¬ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵) ↔ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵))
10	8, 9	bitri 274	. . 3 ⊢ (∀𝑥 ∈ Cℋ ((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝑥 = 𝐵) ↔ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵))
11	10	anbi2i 621	. 2 ⊢ ((𝐴 ⊊ 𝐵 ∧ ∀𝑥 ∈ Cℋ ((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝑥 = 𝐵)) ↔ (𝐴 ⊊ 𝐵 ∧ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵)))
12	1, 11	bitr4di 288	1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∧ ∀𝑥 ∈ Cℋ ((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝑥 = 𝐵))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104  ∀wral 3059  ∃wrex 3068   ⊆ wss 3949   ⊊ wpss 3950   class class class wbr 5149   C_ℋ cch 30447   ⋖_ℋ ccv 30482

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-cv 31797

This theorem is referenced by:  spansncv2  31811  elat2  31858