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Mirrors > Home > MPE Home > Th. List > rextru | Structured version Visualization version GIF version |
Description: Two ways of expressing that a class has at least one element. (Contributed by Zhi Wang, 23-Sep-2024.) |
Ref | Expression |
---|---|
rextru | ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ⊤) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tru 1545 | . . . 4 ⊢ ⊤ | |
2 | 1 | biantru 530 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ ⊤)) |
3 | 2 | exbii 1850 | . 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⊤)) |
4 | df-rex 3071 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ⊤ ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⊤)) | |
5 | 3, 4 | bitr4i 277 | 1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ⊤) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ⊤wtru 1542 ∃wex 1781 ∈ wcel 2106 ∃wrex 3070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-rex 3071 |
This theorem is referenced by: iinss2d 43836 ralfal 43840 reutruALT 47444 |
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