Theorem idssen 9019

Description: Equality implies equinumerosity. (Contributed by NM, 30-Apr-1998.) (Revised by Mario Carneiro, 15-Nov-2014.)

Assertion

Ref	Expression
idssen	⊢ I ⊆ ≈

Proof of Theorem idssen

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

Step	Hyp	Ref	Expression
1		reli 5816	. 2 ⊢ Rel I
2		vex 3467	. . . . 5 ⊢ 𝑦 ∈ V
3	2	ideq 5843	. . . 4 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
4		eqeng 9008	. . . . 5 ⊢ (𝑥 ∈ V → (𝑥 = 𝑦 → 𝑥 ≈ 𝑦))
5	4	elv 3468	. . . 4 ⊢ (𝑥 = 𝑦 → 𝑥 ≈ 𝑦)
6	3, 5	sylbi 217	. . 3 ⊢ (𝑥 I 𝑦 → 𝑥 ≈ 𝑦)
7		df-br 5124	. . 3 ⊢ (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
8		df-br 5124	. . 3 ⊢ (𝑥 ≈ 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ≈ )
9	6, 7, 8	3imtr3i 291	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑦⟩ ∈ ≈ )
10	1, 9	relssi 5777	1 ⊢ I ⊆ ≈

Syntax hints:   → wi 4   ∈ wcel 2107  Vcvv 3463   ⊆ wss 3931  ⟨cop 4612   class class class wbr 5123   I cid 5557   ≈ cen 8964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-en 8968

This theorem is referenced by: (None)