Theorem cvbr2 32891

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 32890	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∧ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵))))
2		iman 407	. . . . . 6 ⊢ (((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝑥 = 𝐵) ↔ ¬ ((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ ¬ 𝑥 = 𝐵))
3		anass 474	. . . . . . 7 ⊢ (((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ ¬ 𝑥 = 𝐵) ↔ (𝐴 ⊊ 𝑥 ∧ (𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 = 𝐵)))
4		dfpss2 4015	. . . . . . . 8 ⊢ (𝑥 ⊊ 𝐵 ↔ (𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 = 𝐵))
5	4	anbi2i 635	. . . . . . 7 ⊢ ((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵) ↔ (𝐴 ⊊ 𝑥 ∧ (𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 = 𝐵)))
6	3, 5	bitr4i 281	. . . . . 6 ⊢ (((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ ¬ 𝑥 = 𝐵) ↔ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵))
7	2, 6	xchbinx 337	. . . . 5 ⊢ (((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝑥 = 𝐵) ↔ ¬ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵))
8	7	ralbii 3108	. . . 4 ⊢ (∀𝑥 ∈ Cℋ ((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝑥 = 𝐵) ↔ ∀𝑥 ∈ Cℋ ¬ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵))
9		ralnex 3088	. . . 4 ⊢ (∀𝑥 ∈ Cℋ ¬ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵) ↔ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵))
10	8, 9	bitri 278	. . 3 ⊢ (∀𝑥 ∈ Cℋ ((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝑥 = 𝐵) ↔ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵))
11	10	anbi2i 635	. 2 ⊢ ((𝐴 ⊊ 𝐵 ∧ ∀𝑥 ∈ Cℋ ((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝑥 = 𝐵)) ↔ (𝐴 ⊊ 𝐵 ∧ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵)))
12	1, 11	bitr4di 292	1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∧ ∀𝑥 ∈ Cℋ ((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝑥 = 𝐵))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086   ⊆ wss 3895   ⊊ wpss 4013   class class class wbr 5102   C_ℋ cch 31537   ⋖_ℋ ccv 31572

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-cv 32887

This theorem is used by:  spansncv2  32901  elat2  32948