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Mirrors > Home > ILE Home > Th. List > dfss2 | GIF version |
Description: Alternate definition of the subclass relationship between two classes. Definition 5.9 of [TakeutiZaring] p. 17. (Contributed by NM, 8-Jan-2002.) |
Ref | Expression |
---|---|
dfss2 | ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfss 3126 | . . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 = (𝐴 ∩ 𝐵)) | |
2 | df-in 3118 | . . . 4 ⊢ (𝐴 ∩ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} | |
3 | 2 | eqeq2i 2175 | . . 3 ⊢ (𝐴 = (𝐴 ∩ 𝐵) ↔ 𝐴 = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}) |
4 | abeq2 2273 | . . 3 ⊢ (𝐴 = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))) | |
5 | 1, 3, 4 | 3bitri 205 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))) |
6 | pm4.71 387 | . . 3 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))) | |
7 | 6 | albii 1457 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))) |
8 | 5, 7 | bitr4i 186 | 1 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∀wal 1340 = wceq 1342 ∈ wcel 2135 {cab 2150 ∩ cin 3111 ⊆ wss 3112 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-11 1493 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-in 3118 df-ss 3125 |
This theorem is referenced by: dfss3 3128 dfss2f 3129 ssel 3132 ssriv 3142 ssrdv 3144 sstr2 3145 eqss 3153 nssr 3198 rabss2 3221 ssconb 3251 ssequn1 3288 unss 3292 ssin 3340 ssddif 3352 reldisj 3456 ssdif0im 3469 inssdif0im 3472 ssundifim 3488 sbcssg 3514 pwss 3570 snss 3697 snsssn 3736 ssuni 3806 unissb 3814 intss 3840 iunss 3902 dftr2 4077 axpweq 4145 axpow2 4150 ssextss 4193 ordunisuc2r 4486 setind 4511 zfregfr 4546 tfi 4554 ssrel 4687 ssrel2 4689 ssrelrel 4699 reliun 4720 relop 4749 issref 4981 funimass4 5532 isprm2 12038 bj-inf2vnlem3 13716 bj-inf2vnlem4 13717 |
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