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Theorem idssen 7945
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 5214 . 2 Rel I
2 vex 3194 . . . . 5 𝑦 ∈ V
32ideq 5239 . . . 4 (𝑥 I 𝑦𝑥 = 𝑦)
4 vex 3194 . . . . 5 𝑥 ∈ V
5 eqeng 7934 . . . . 5 (𝑥 ∈ V → (𝑥 = 𝑦𝑥𝑦))
64, 5ax-mp 5 . . . 4 (𝑥 = 𝑦𝑥𝑦)
73, 6sylbi 207 . . 3 (𝑥 I 𝑦𝑥𝑦)
8 df-br 4619 . . 3 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
9 df-br 4619 . . 3 (𝑥𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ≈ )
107, 8, 93imtr3i 280 . 2 (⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑦⟩ ∈ ≈ )
111, 10relssi 5177 1 I ⊆ ≈
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1992  Vcvv 3191  wss 3560  cop 4159   class class class wbr 4618   I cid 4989  cen 7897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-en 7901
This theorem is referenced by: (None)
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