Theorem brcnvssrid 39722

Description: Any set is a converse subset of itself. (Contributed by Peter Mazsa, 9-Jun-2021.)

Assertion

Ref	Expression
brcnvssrid	⊢ (𝐴 ∈ 𝑉 → 𝐴◡ S 𝐴)

Proof of Theorem brcnvssrid

Step	Hyp	Ref	Expression
1		ssid 3935	. 2 ⊢ 𝐴 ⊆ 𝐴
2		brcnvssr 39721	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴◡ S 𝐴 ↔ 𝐴 ⊆ 𝐴))
3	1, 2	mpbiri 261	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴◡ S 𝐴)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147   ⊆ wss 3900   class class class wbr 5106  ^◡ccnv 5655   S cssr 39321

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5253  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5172  df-xp 5662  df-rel 5663  df-cnv 5664  df-ssr 39713

This theorem is used by: (None)