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Theorem idssen 8175
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 5400 . 2 Rel I
2 vex 3358 . . . . 5 𝑦 ∈ V
32ideq 5425 . . . 4 (𝑥 I 𝑦𝑥 = 𝑦)
4 vex 3358 . . . . 5 𝑥 ∈ V
5 eqeng 8164 . . . . 5 (𝑥 ∈ V → (𝑥 = 𝑦𝑥𝑦))
64, 5ax-mp 5 . . . 4 (𝑥 = 𝑦𝑥𝑦)
73, 6sylbi 208 . . 3 (𝑥 I 𝑦𝑥𝑦)
8 df-br 4798 . . 3 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
9 df-br 4798 . . 3 (𝑥𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ≈ )
107, 8, 93imtr3i 281 . 2 (⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑦⟩ ∈ ≈ )
111, 10relssi 5363 1 I ⊆ ≈
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2148  Vcvv 3355  wss 3729  cop 4332   class class class wbr 4797   I cid 5170  cen 8127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-8 2150  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-sep 4928  ax-nul 4936  ax-pow 4988  ax-pr 5048  ax-un 7117
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3an 1100  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3357  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-nul 4074  df-if 4236  df-pw 4309  df-sn 4327  df-pr 4329  df-op 4333  df-uni 4586  df-br 4798  df-opab 4860  df-id 5171  df-xp 5269  df-rel 5270  df-cnv 5271  df-co 5272  df-dm 5273  df-rn 5274  df-res 5275  df-ima 5276  df-fun 6044  df-fn 6045  df-f 6046  df-f1 6047  df-fo 6048  df-f1o 6049  df-en 8131
This theorem is referenced by: (None)
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