Theorem idssen 6625

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 4628	. 2 ⊢ Rel I
2		vex 2660	. . . . 5 ⊢ 𝑦 ∈ V
3	2	ideq 4651	. . . 4 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
4		vex 2660	. . . . 5 ⊢ 𝑥 ∈ V
5		eqeng 6614	. . . . 5 ⊢ (𝑥 ∈ V → (𝑥 = 𝑦 → 𝑥 ≈ 𝑦))
6	4, 5	ax-mp 7	. . . 4 ⊢ (𝑥 = 𝑦 → 𝑥 ≈ 𝑦)
7	3, 6	sylbi 120	. . 3 ⊢ (𝑥 I 𝑦 → 𝑥 ≈ 𝑦)
8		df-br 3896	. . 3 ⊢ (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
9		df-br 3896	. . 3 ⊢ (𝑥 ≈ 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ≈ )
10	7, 8, 9	3imtr3i 199	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑦⟩ ∈ ≈ )
11	1, 10	relssi 4590	1 ⊢ I ⊆ ≈

Syntax hints:   → wi 4   ∈ wcel 1463  Vcvv 2657   ⊆ wss 3037  ⟨cop 3496   class class class wbr 3895   I cid 4170   ≈ cen 6586

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-opab 3950  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-en 6589

This theorem is referenced by: (None)