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Mirrors > Home > ILE Home > Th. List > dfss2f | GIF version |
Description: Equivalence for subclass relation, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 3-Jul-1994.) (Revised by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
dfss2f.1 | ⊢ Ⅎ𝑥𝐴 |
dfss2f.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
dfss2f | ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfss2 3081 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑧(𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵)) | |
2 | dfss2f.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | nfcri 2273 | . . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴 |
4 | dfss2f.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
5 | 4 | nfcri 2273 | . . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐵 |
6 | 3, 5 | nfim 1551 | . . 3 ⊢ Ⅎ𝑥(𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵) |
7 | nfv 1508 | . . 3 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) | |
8 | eleq1 2200 | . . . 4 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)) | |
9 | eleq1 2200 | . . . 4 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝐵 ↔ 𝑥 ∈ 𝐵)) | |
10 | 8, 9 | imbi12d 233 | . . 3 ⊢ (𝑧 = 𝑥 → ((𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))) |
11 | 6, 7, 10 | cbval 1727 | . 2 ⊢ (∀𝑧(𝑧 ∈ 𝐴 → 𝑧 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
12 | 1, 11 | bitri 183 | 1 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1329 ∈ wcel 1480 Ⅎwnfc 2266 ⊆ wss 3066 |
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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-in 3072 df-ss 3079 |
This theorem is referenced by: dfss3f 3084 ssrd 3097 ssrmof 3155 ss2ab 3160 |
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