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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 1416 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ↔ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ∧ ∀𝑥(𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴))) | |
2 | dfcleq 2075 | . 2 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | |
3 | dfss2 2988 | . . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | |
4 | dfss2 2988 | . . 3 ⊢ (𝐵 ⊆ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴)) | |
5 | 3, 4 | anbi12i 447 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴) ↔ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ∧ ∀𝑥(𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴))) |
6 | 1, 2, 5 | 3bitr4i 210 | 1 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∀wal 1282 = wceq 1284 ∈ wcel 1433 ⊆ wss 2973 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-11 1437 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-in 2979 df-ss 2986 |
This theorem is referenced by: eqssi 3015 eqssd 3016 sseq1 3020 sseq2 3021 eqimss 3051 ssrabeq 3080 uneqin 3215 ss0b 3283 vss 3291 sssnm 3546 unidif 3633 ssunieq 3634 iuneq1 3691 iuneq2 3694 iunxdif2 3726 ssext 3976 pweqb 3978 eqopab2b 4034 pwunim 4041 soeq2 4071 iunpw 4229 ordunisuc2r 4258 tfi 4323 eqrel 4447 eqrelrel 4459 coeq1 4511 coeq2 4512 cnveq 4527 dmeq 4553 relssres 4666 xp11m 4779 xpcanm 4780 xpcan2m 4781 ssrnres 4783 fnres 5035 eqfnfv3 5288 fneqeql2 5297 fconst4m 5402 f1imaeq 5435 eqoprab2b 5583 fo1stresm 5808 fo2ndresm 5809 nnacan 6108 nnmcan 6115 isprm2 10499 bj-sseq 10602 bdeq0 10658 bdvsn 10665 bdop 10666 bdeqsuc 10672 bj-om 10732 |
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