Theorem idssen 6831

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 4791	. 2 ⊢ Rel I
2		vex 2763	. . . . 5 ⊢ 𝑦 ∈ V
3	2	ideq 4814	. . . 4 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
4		vex 2763	. . . . 5 ⊢ 𝑥 ∈ V
5		eqeng 6820	. . . . 5 ⊢ (𝑥 ∈ V → (𝑥 = 𝑦 → 𝑥 ≈ 𝑦))
6	4, 5	ax-mp 5	. . . 4 ⊢ (𝑥 = 𝑦 → 𝑥 ≈ 𝑦)
7	3, 6	sylbi 121	. . 3 ⊢ (𝑥 I 𝑦 → 𝑥 ≈ 𝑦)
8		df-br 4030	. . 3 ⊢ (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
9		df-br 4030	. . 3 ⊢ (𝑥 ≈ 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ≈ )
10	7, 8, 9	3imtr3i 200	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑦⟩ ∈ ≈ )
11	1, 10	relssi 4750	1 ⊢ I ⊆ ≈

Syntax hints:   → wi 4   ∈ wcel 2164  Vcvv 2760   ⊆ wss 3153  ⟨cop 3621   class class class wbr 4029   I cid 4319   ≈ cen 6792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-en 6795

This theorem is referenced by: (None)