![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > eqss | GIF version |
Description: The subclass relationship is antisymmetric. Compare Theorem 4 of [Suppes] p. 22. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
eqss | ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | albiim 1376 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ↔ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ∧ ∀𝑥(𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴))) | |
2 | dfcleq 2034 | . 2 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | |
3 | dfss2 2934 | . . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | |
4 | dfss2 2934 | . . 3 ⊢ (𝐵 ⊆ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴)) | |
5 | 3, 4 | anbi12i 433 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴) ↔ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ∧ ∀𝑥(𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴))) |
6 | 1, 2, 5 | 3bitr4i 201 | 1 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∀wal 1241 = wceq 1243 ∈ wcel 1393 ⊆ wss 2917 |
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-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-11 1397 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-in 2924 df-ss 2931 |
This theorem is referenced by: eqssi 2961 eqssd 2962 sseq1 2966 sseq2 2967 eqimss 2997 ssrabeq 3026 dfpss3 3030 uneqin 3188 ss0b 3256 vss 3264 sssnm 3525 unidif 3612 ssunieq 3613 iuneq1 3670 iuneq2 3673 iunxdif2 3705 ssext 3957 pweqb 3959 eqopab2b 4016 pwunim 4023 soeq2 4053 iunpw 4211 ordunisuc2r 4240 tfi 4305 eqrel 4429 eqrelrel 4441 coeq1 4493 coeq2 4494 cnveq 4509 dmeq 4535 relssres 4648 xp11m 4759 xpcanm 4760 xpcan2m 4761 ssrnres 4763 fnres 5015 eqfnfv3 5267 fneqeql2 5276 fconst4m 5381 f1imaeq 5414 eqoprab2b 5563 fo1stresm 5788 fo2ndresm 5789 nnacan 6085 nnmcan 6092 bj-sseq 9931 bdeq0 9987 bdvsn 9994 bdop 9995 bdeqsuc 10001 bj-om 10061 |
Copyright terms: Public domain | W3C validator |