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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 114 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴)) |
4 | 3 | ssrdv 3108 | . 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
5 | 1, 4 | eqssd 3119 | 1 ⊢ (𝜑 → 𝐴 = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1332 ∈ wcel 1481 ⊆ wss 3076 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-11 1485 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-in 3082 df-ss 3089 |
This theorem is referenced by: fiuni 6874 ennnfonelemrn 11968 ennnfonelemdm 11969 unirnblps 12630 unirnbl 12631 dvidlemap 12868 dviaddf 12877 dvimulf 12878 dvcj 12881 dvrecap 12885 |
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