Theorem idssen 8537

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 5662	. 2 ⊢ Rel I
2		vex 3444	. . . . 5 ⊢ 𝑦 ∈ V
3	2	ideq 5687	. . . 4 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
4		eqeng 8526	. . . . 5 ⊢ (𝑥 ∈ V → (𝑥 = 𝑦 → 𝑥 ≈ 𝑦))
5	4	elv 3446	. . . 4 ⊢ (𝑥 = 𝑦 → 𝑥 ≈ 𝑦)
6	3, 5	sylbi 220	. . 3 ⊢ (𝑥 I 𝑦 → 𝑥 ≈ 𝑦)
7		df-br 5031	. . 3 ⊢ (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
8		df-br 5031	. . 3 ⊢ (𝑥 ≈ 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ≈ )
9	6, 7, 8	3imtr3i 294	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑦⟩ ∈ ≈ )
10	1, 9	relssi 5624	1 ⊢ I ⊆ ≈

Syntax hints:   → wi 4   ∈ wcel 2111  Vcvv 3441   ⊆ wss 3881  ⟨cop 4531   class class class wbr 5030   I cid 5424   ≈ cen 8489

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-en 8493

This theorem is referenced by: (None)