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| Mirrors > Home > ILE Home > Th. List > eqelssd | GIF version | ||
| Description: Equality deduction from subclass relationship and membership. (Contributed by AV, 21-Aug-2022.) |
| Ref | Expression |
|---|---|
| eqelssd.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
| eqelssd.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐴) |
| Ref | Expression |
|---|---|
| eqelssd | ⊢ (𝜑 → 𝐴 = 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqelssd.1 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
| 2 | eqelssd.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐴) | |
| 3 | 2 | ex 115 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴)) |
| 4 | 3 | ssrdv 3199 | . 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
| 5 | 1, 4 | eqssd 3210 | 1 ⊢ (𝜑 → 𝐴 = 𝐵) |
| Colors of variables: wff set class |
| Syntax hints: → wi 4 ∧ wa 104 = wceq 1373 ∈ wcel 2176 ⊆ wss 3166 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-11 1529 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-ext 2187 |
| This theorem depends on definitions: df-bi 117 df-nf 1484 df-sb 1786 df-clab 2192 df-cleq 2198 df-clel 2201 df-in 3172 df-ss 3179 |
| This theorem is referenced by: fiuni 7080 ennnfonelemrn 12790 ennnfonelemdm 12791 unirnblps 14894 unirnbl 14895 dvidlemap 15163 dvidrelem 15164 dvidsslem 15165 dviaddf 15177 dvimulf 15178 dvcj 15181 dvrecap 15185 |
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