Theorem cvpss 32229

Description: The covers relation implies proper subset. (Contributed by NM, 10-Jun-2004.) (New usage is discouraged.)

Assertion

Ref	Expression
cvpss	⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → 𝐴 ⊊ 𝐵))

Proof of Theorem cvpss

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cvbr 32226	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∧ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵))))
2		simpl 482	. 2 ⊢ ((𝐴 ⊊ 𝐵 ∧ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵)) → 𝐴 ⊊ 𝐵)
3	1, 2	biimtrdi 253	1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → 𝐴 ⊊ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∈ wcel 2109  ∃wrex 3053   ⊊ wpss 3904   class class class wbr 5092   C_ℋ cch 30873   ⋖_ℋ ccv 30908

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

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-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5093  df-opab 5155  df-cv 32223

This theorem is referenced by:  cvnsym  32234  cvntr  32236  atcveq0  32292  chcv1  32299  cvati  32310  cvbr4i  32311  cvexchlem  32312  atexch  32325  atcvat2i  32331