Theorem extid 34081

Description: Property of identity relation, cf. extep 34048, ~? extssr and the comment of ~? df-ssr . (Contributed by Peter Mazsa, 5-Jul-2019.)

Assertion

Ref	Expression
extid	⊢ (𝐴 ∈ 𝑉 → ([𝐴]◡ I = [𝐵]◡ I ↔ 𝐴 = 𝐵))

Proof of Theorem extid

Step	Hyp	Ref	Expression
1		cnvi 5537	. . . . 5 ⊢ ◡ I = I
2	1	eceq2i 34040	. . . 4 ⊢ [𝐴]◡ I = [𝐴] I
3		ecidsn 7795	. . . 4 ⊢ [𝐴] I = {𝐴}
4	2, 3	eqtri 2644	. . 3 ⊢ [𝐴]◡ I = {𝐴}
5	1	eceq2i 34040	. . . 4 ⊢ [𝐵]◡ I = [𝐵] I
6		ecidsn 7795	. . . 4 ⊢ [𝐵] I = {𝐵}
7	5, 6	eqtri 2644	. . 3 ⊢ [𝐵]◡ I = {𝐵}
8	4, 7	eqeq12i 2636	. 2 ⊢ ([𝐴]◡ I = [𝐵]◡ I ↔ {𝐴} = {𝐵})
9		sneqbg 4374	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵))
10	8, 9	syl5bb 272	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴]◡ I = [𝐵]◡ I ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1483   ∈ wcel 1990  {csn 4177   I cid 5023  ^◡ccnv 5113  [cec 7740

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ec 7744

This theorem is referenced by: (None)