Theorem cvbr2 32311

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 32310	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∧ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵))))
2		iman 401	. . . . . 6 ⊢ (((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝑥 = 𝐵) ↔ ¬ ((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ ¬ 𝑥 = 𝐵))
3		anass 468	. . . . . . 7 ⊢ (((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ ¬ 𝑥 = 𝐵) ↔ (𝐴 ⊊ 𝑥 ∧ (𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 = 𝐵)))
4		dfpss2 4097	. . . . . . . 8 ⊢ (𝑥 ⊊ 𝐵 ↔ (𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 = 𝐵))
5	4	anbi2i 623	. . . . . . 7 ⊢ ((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵) ↔ (𝐴 ⊊ 𝑥 ∧ (𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 = 𝐵)))
6	3, 5	bitr4i 278	. . . . . 6 ⊢ (((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ ¬ 𝑥 = 𝐵) ↔ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵))
7	2, 6	xchbinx 334	. . . . 5 ⊢ (((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝑥 = 𝐵) ↔ ¬ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵))
8	7	ralbii 3090	. . . 4 ⊢ (∀𝑥 ∈ Cℋ ((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝑥 = 𝐵) ↔ ∀𝑥 ∈ Cℋ ¬ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵))
9		ralnex 3069	. . . 4 ⊢ (∀𝑥 ∈ Cℋ ¬ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵) ↔ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵))
10	8, 9	bitri 275	. . 3 ⊢ (∀𝑥 ∈ Cℋ ((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝑥 = 𝐵) ↔ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵))
11	10	anbi2i 623	. 2 ⊢ ((𝐴 ⊊ 𝐵 ∧ ∀𝑥 ∈ Cℋ ((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝑥 = 𝐵)) ↔ (𝐴 ⊊ 𝐵 ∧ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵)))
12	1, 11	bitr4di 289	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 206   ∧ wa 395   = wceq 1536   ∈ wcel 2105  ∀wral 3058  ∃wrex 3067   ⊆ wss 3962   ⊊ wpss 3963   class class class wbr 5147   C_ℋ cch 30957   ⋖_ℋ ccv 30992

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-cv 32307

This theorem is referenced by:  spansncv2  32321  elat2  32368