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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-axemptylem | GIF version |
Description: Lemma for bj-axempty 13262 and bj-axempty2 13263. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4062 instead. (New usage is discouraged.) |
Ref | Expression |
---|---|
bj-axemptylem | ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 → ⊥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bdfal 13202 | . . 3 ⊢ BOUNDED ⊥ | |
2 | 1 | bdsep1 13254 | . 2 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ ⊥)) |
3 | bi1 117 | . . . 4 ⊢ ((𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ ⊥)) → (𝑦 ∈ 𝑥 → (𝑦 ∈ 𝑧 ∧ ⊥))) | |
4 | falimd 1347 | . . . 4 ⊢ ((𝑦 ∈ 𝑧 ∧ ⊥) → ⊥) | |
5 | 3, 4 | syl6 33 | . . 3 ⊢ ((𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ ⊥)) → (𝑦 ∈ 𝑥 → ⊥)) |
6 | 5 | alimi 1432 | . 2 ⊢ (∀𝑦(𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑧 ∧ ⊥)) → ∀𝑦(𝑦 ∈ 𝑥 → ⊥)) |
7 | 2, 6 | eximii 1582 | 1 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 → ⊥) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∀wal 1330 ⊥wfal 1337 ∃wex 1469 |
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-in1 604 ax-in2 605 ax-5 1424 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-4 1488 ax-ial 1515 ax-bd0 13182 ax-bdim 13183 ax-bdn 13186 ax-bdeq 13189 ax-bdsep 13253 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-fal 1338 |
This theorem is referenced by: bj-axempty 13262 bj-axempty2 13263 |
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