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Mirrors > Home > NFE Home > Th. List > eqrel | GIF version |
Description: Extensionality principle for relations. Theorem 3.2(ii) of [Monk1] p. 33. (Contributed by NM, 2-Aug-1994.) (Revised by Scott Fenton, 14-Apr-2021.) |
Ref | Expression |
---|---|
eqrel | ⊢ (A = B ↔ ∀x∀y(〈x, y〉 ∈ A ↔ 〈x, y〉 ∈ B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrel 4844 | . . 3 ⊢ (A ⊆ B ↔ ∀x∀y(〈x, y〉 ∈ A → 〈x, y〉 ∈ B)) | |
2 | ssrel 4844 | . . 3 ⊢ (B ⊆ A ↔ ∀x∀y(〈x, y〉 ∈ B → 〈x, y〉 ∈ A)) | |
3 | 1, 2 | anbi12i 678 | . 2 ⊢ ((A ⊆ B ∧ B ⊆ A) ↔ (∀x∀y(〈x, y〉 ∈ A → 〈x, y〉 ∈ B) ∧ ∀x∀y(〈x, y〉 ∈ B → 〈x, y〉 ∈ A))) |
4 | eqss 3287 | . 2 ⊢ (A = B ↔ (A ⊆ B ∧ B ⊆ A)) | |
5 | 2albiim 1612 | . 2 ⊢ (∀x∀y(〈x, y〉 ∈ A ↔ 〈x, y〉 ∈ B) ↔ (∀x∀y(〈x, y〉 ∈ A → 〈x, y〉 ∈ B) ∧ ∀x∀y(〈x, y〉 ∈ B → 〈x, y〉 ∈ A))) | |
6 | 3, 4, 5 | 3bitr4i 268 | 1 ⊢ (A = B ↔ ∀x∀y(〈x, y〉 ∈ A ↔ 〈x, y〉 ∈ B)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 ∀wal 1540 = wceq 1642 ∈ wcel 1710 ⊆ wss 3257 〈cop 4561 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-0c 4377 df-addc 4378 df-nnc 4379 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 |
This theorem is referenced by: eqrelriv 4850 eqrelrdv 4852 dmeq0 4922 eqfnfv 5392 |
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