Theorem enssdom 8969

Description: Equinumerosity implies dominance. (Contributed by NM, 31-Mar-1998.) (Proof shortened by TM, 10-Feb-2026.)

Assertion

Ref	Expression
enssdom	⊢ ≈ ⊆ ≼

Proof of Theorem enssdom

Dummy variables 𝑥 𝑦 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1of1 6817	. . . 4 ⊢ (𝑓:𝑥–1-1-onto→𝑦 → 𝑓:𝑥–1-1→𝑦)
2	1	eximi 1862	. . 3 ⊢ (∃𝑓 𝑓:𝑥–1-1-onto→𝑦 → ∃𝑓 𝑓:𝑥–1-1→𝑦)
3	2	ssopab2i 5533	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦} ⊆ {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦}
4		df-en 8940	. 2 ⊢ ≈ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦}
5		df-dom 8941	. 2 ⊢ ≼ = {⟨𝑥, 𝑦⟩ ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦}
6	3, 4, 5	3sstr4i 3996	1 ⊢ ≈ ⊆ ≼

Syntax hints:  ∃wex 1806   ⊆ wss 3913  {copab 5174  –1-1→wf1 6531  –1-1-onto→wf1o 6533   ≈ cen 8936   ≼ cdom 8937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-ss 3930  df-opab 5175  df-f1o 6541  df-en 8940  df-dom 8941

This theorem is referenced by:  dfdom2  8971  endom  8972