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Theorem idssen 6257
 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 4465 . 2 Rel I
2 vex 2560 . . . . 5 𝑦 ∈ V
32ideq 4488 . . . 4 (𝑥 I 𝑦𝑥 = 𝑦)
4 vex 2560 . . . . 5 𝑥 ∈ V
5 eqeng 6246 . . . . 5 (𝑥 ∈ V → (𝑥 = 𝑦𝑥𝑦))
64, 5ax-mp 7 . . . 4 (𝑥 = 𝑦𝑥𝑦)
73, 6sylbi 114 . . 3 (𝑥 I 𝑦𝑥𝑦)
8 df-br 3765 . . 3 (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
9 df-br 3765 . . 3 (𝑥𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ≈ )
107, 8, 93imtr3i 189 . 2 (⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑦⟩ ∈ ≈ )
111, 10relssi 4431 1 I ⊆ ≈
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1393  Vcvv 2557   ⊆ wss 2917  ⟨cop 3378   class class class wbr 3764   I cid 4025   ≈ cen 6219 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-en 6222 This theorem is referenced by: (None)
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