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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 1467 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) ↔ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ∧ ∀𝑥(𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴))) | |
2 | dfcleq 2151 | . 2 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | |
3 | dfss2 3117 | . . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | |
4 | dfss2 3117 | . . 3 ⊢ (𝐵 ⊆ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴)) | |
5 | 3, 4 | anbi12i 456 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴) ↔ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ∧ ∀𝑥(𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴))) |
6 | 1, 2, 5 | 3bitr4i 211 | 1 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∀wal 1333 = wceq 1335 ∈ wcel 2128 ⊆ wss 3102 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-11 1486 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-in 3108 df-ss 3115 |
This theorem is referenced by: eqssi 3144 eqssd 3145 sseq1 3151 sseq2 3152 eqimss 3182 ssrabeq 3215 uneqin 3359 ss0b 3434 vss 3442 sssnm 3719 unidif 3806 ssunieq 3807 iuneq1 3864 iuneq2 3867 iunxdif2 3899 ssext 4183 pweqb 4185 eqopab2b 4241 pwunim 4248 soeq2 4278 iunpw 4442 ordunisuc2r 4475 tfi 4543 eqrel 4677 eqrelrel 4689 coeq1 4745 coeq2 4746 cnveq 4762 dmeq 4788 relssres 4906 xp11m 5026 xpcanm 5027 xpcan2m 5028 ssrnres 5030 fnres 5288 eqfnfv3 5569 fneqeql2 5578 fconst4m 5689 f1imaeq 5727 eqoprab2b 5881 fo1stresm 6111 fo2ndresm 6112 nnacan 6461 nnmcan 6468 ixpeq2 6659 sbthlemi3 6905 isprm2 12009 bastop1 12553 epttop 12560 opnneiid 12634 cnntr 12695 metequiv 12965 bj-sseq 13437 bdeq0 13513 bdvsn 13520 bdop 13521 bdeqsuc 13527 bj-om 13583 |
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