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Theorem idssen 6639
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.
StepHypRef Expression
1 reli 4638 . 2 Rel I
2 vex 2663 . . . . 5 𝑦 ∈ V
32ideq 4661 . . . 4 (𝑥 I 𝑦𝑥 = 𝑦)
4 vex 2663 . . . . 5 𝑥 ∈ V
5 eqeng 6628 . . . . 5 (𝑥 ∈ V → (𝑥 = 𝑦𝑥𝑦))
64, 5ax-mp 5 . . . 4 (𝑥 = 𝑦𝑥𝑦)
73, 6sylbi 120 . . 3 (𝑥 I 𝑦𝑥𝑦)
8 df-br 3900 . . 3 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
9 df-br 3900 . . 3 (𝑥𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ≈ )
107, 8, 93imtr3i 199 . 2 (⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑦⟩ ∈ ≈ )
111, 10relssi 4600 1 I ⊆ ≈
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1465  Vcvv 2660  wss 3041  cop 3500   class class class wbr 3899   I cid 4180  cen 6600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-en 6603
This theorem is referenced by: (None)
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