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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 3085 | . . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐴 = (𝐴 ∩ 𝐵)) | |
2 | df-in 3077 | . . . 4 ⊢ (𝐴 ∩ 𝐵) = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} | |
3 | 2 | eqeq2i 2150 | . . 3 ⊢ (𝐴 = (𝐴 ∩ 𝐵) ↔ 𝐴 = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)}) |
4 | abeq2 2248 | . . 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 | pm4.71 386 | . . 3 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))) | |
7 | 6 | albii 1446 | . 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 1329 = wceq 1331 ∈ wcel 1480 {cab 2125 ∩ cin 3070 ⊆ wss 3071 |
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 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-11 1484 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-in 3077 df-ss 3084 |
This theorem is referenced by: dfss3 3087 dfss2f 3088 ssel 3091 ssriv 3101 ssrdv 3103 sstr2 3104 eqss 3112 nssr 3157 rabss2 3180 ssconb 3209 ssequn1 3246 unss 3250 ssin 3298 ssddif 3310 reldisj 3414 ssdif0im 3427 inssdif0im 3430 ssundifim 3446 sbcssg 3472 pwss 3526 snss 3649 snsssn 3688 ssuni 3758 unissb 3766 intss 3792 iunss 3854 dftr2 4028 axpweq 4095 axpow2 4100 ssextss 4142 ordunisuc2r 4430 setind 4454 zfregfr 4488 tfi 4496 ssrel 4627 ssrel2 4629 ssrelrel 4639 reliun 4660 relop 4689 issref 4921 funimass4 5472 isprm2 11798 bj-inf2vnlem3 13170 bj-inf2vnlem4 13171 |
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