cvbr2 - Hilbert Space Explorer

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Theorem cvbr2 30062

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 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ) → (𝐴 ⋖_ℋ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∧ ∀𝑥 ∈ C_ℋ ((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝑥 = 𝐵))))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵

**Proof of Theorem cvbr2**
Step	Hyp	Ref	Expression
1		cvbr 30061	. 2 ⊢ ((𝐴 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ) → (𝐴 ⋖_ℋ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∧ ¬ ∃𝑥 ∈ C_ℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵))))
2		iman 404	. . . . . 6 ⊢ (((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝑥 = 𝐵) ↔ ¬ ((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ ¬ 𝑥 = 𝐵))
3		anass 471	. . . . . . 7 ⊢ (((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ ¬ 𝑥 = 𝐵) ↔ (𝐴 ⊊ 𝑥 ∧ (𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 = 𝐵)))
4		dfpss2 4064	. . . . . . . 8 ⊢ (𝑥 ⊊ 𝐵 ↔ (𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 = 𝐵))
5	4	anbi2i 624	. . . . . . 7 ⊢ ((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵) ↔ (𝐴 ⊊ 𝑥 ∧ (𝑥 ⊆ 𝐵 ∧ ¬ 𝑥 = 𝐵)))
6	3, 5	bitr4i 280	. . . . . 6 ⊢ (((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) ∧ ¬ 𝑥 = 𝐵) ↔ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵))
7	2, 6	xchbinx 336	. . . . 5 ⊢ (((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝑥 = 𝐵) ↔ ¬ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵))
8	7	ralbii 3167	. . . 4 ⊢ (∀𝑥 ∈ C_ℋ ((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝑥 = 𝐵) ↔ ∀𝑥 ∈ C_ℋ ¬ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵))
9		ralnex 3238	. . . 4 ⊢ (∀𝑥 ∈ C_ℋ ¬ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵) ↔ ¬ ∃𝑥 ∈ C_ℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵))
10	8, 9	bitri 277	. . 3 ⊢ (∀𝑥 ∈ C_ℋ ((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝑥 = 𝐵) ↔ ¬ ∃𝑥 ∈ C_ℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵))
11	10	anbi2i 624	. 2 ⊢ ((𝐴 ⊊ 𝐵 ∧ ∀𝑥 ∈ C_ℋ ((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝑥 = 𝐵)) ↔ (𝐴 ⊊ 𝐵 ∧ ¬ ∃𝑥 ∈ C_ℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵)))
12	1, 11	syl6bbr 291	1 ⊢ ((𝐴 ∈ C_ℋ ∧ 𝐵 ∈ C_ℋ ) → (𝐴 ⋖_ℋ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∧ ∀𝑥 ∈ C_ℋ ((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊆ 𝐵) → 𝑥 = 𝐵))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∃wrex 3141 ⊆ wss 3938 ⊊ wpss 3939 class class class wbr 5068 C_ℋ cch 28708 ⋖_ℋ ccv 28743

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-cv 30058

This theorem is referenced by: spansncv2 30072 elat2 30119

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