Theorem 0domg 9044

Description: Any set dominates the empty set. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.) Avoid ax-pow 5312, ax-un 7690. (Revised by BTernaryTau, 29-Nov-2024.)

Assertion

Ref	Expression
0domg	⊢ (𝐴 ∈ 𝑉 → ∅ ≼ 𝐴)

Proof of Theorem 0domg

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 5254	. . 3 ⊢ ∅ ∈ V
2		f1eq1 6733	. . 3 ⊢ (𝑓 = ∅ → (𝑓:∅–1-1→𝐴 ↔ ∅:∅–1-1→𝐴))
3		f10 6815	. . 3 ⊢ ∅:∅–1-1→𝐴
4	1, 2, 3	ceqsexv2d 3493	. 2 ⊢ ∃𝑓 𝑓:∅–1-1→𝐴
5		brdom2g 8906	. . 3 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ 𝑉) → (∅ ≼ 𝐴 ↔ ∃𝑓 𝑓:∅–1-1→𝐴))
6	1, 5	mpan 691	. 2 ⊢ (𝐴 ∈ 𝑉 → (∅ ≼ 𝐴 ↔ ∃𝑓 𝑓:∅–1-1→𝐴))
7	4, 6	mpbiri 258	1 ⊢ (𝐴 ∈ 𝑉 → ∅ ≼ 𝐴)

Syntax hints:   → wi 4   ↔ wb 206  ∃wex 1781   ∈ wcel 2114  Vcvv 3442  ∅c0 4287   class class class wbr 5100  –1-1→wf1 6497   ≼ cdom 8893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2540  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-dom 8897

This theorem is referenced by:  0sdomg  9046  0dom  9047  carddomi2  9894  wdomfil  9983  wdomnumr  9986  hashge0  14322  ufildom1  23882  harn0  43459  safesnsupfidom1o  43773  sn1dom  43882