Theorem idssen 8934

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 5769	. 2 ⊢ Rel I
2		vex 3435	. . . . 5 ⊢ 𝑦 ∈ V
3	2	ideq 5794	. . . 4 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
4		eqeng 8923	. . . . 5 ⊢ (𝑥 ∈ V → (𝑥 = 𝑦 → 𝑥 ≈ 𝑦))
5	4	elv 3436	. . . 4 ⊢ (𝑥 = 𝑦 → 𝑥 ≈ 𝑦)
6	3, 5	sylbi 218	. . 3 ⊢ (𝑥 I 𝑦 → 𝑥 ≈ 𝑦)
7		df-br 5073	. . 3 ⊢ (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
8		df-br 5073	. . 3 ⊢ (𝑥 ≈ 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ≈ )
9	6, 7, 8	3imtr3i 292	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑦⟩ ∈ ≈ )
10	1, 9	relssi 5730	1 ⊢ I ⊆ ≈

Syntax hints:   → wi 4   ∈ wcel 2119  Vcvv 3431   ⊆ wss 3883  ⟨cop 4561   class class class wbr 5072   I cid 5512   ≈ cen 8880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-en 8884

This theorem is referenced by: (None)