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Mirrors > Home > MPE Home > Th. List > Mathboxes > reutruALT | Structured version Visualization version GIF version |
Description: Alternate proof for reutru 45767. (Contributed by Zhi Wang, 23-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
reutruALT | ⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃!𝑥 ∈ 𝐴 ⊤) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rextru 45765 | . . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ⊤) | |
2 | rmotru 45766 | . . 3 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃*𝑥 ∈ 𝐴 ⊤) | |
3 | 1, 2 | anbi12i 630 | . 2 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∃*𝑥 𝑥 ∈ 𝐴) ↔ (∃𝑥 ∈ 𝐴 ⊤ ∧ ∃*𝑥 ∈ 𝐴 ⊤)) |
4 | df-eu 2568 | . 2 ⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ (∃𝑥 𝑥 ∈ 𝐴 ∧ ∃*𝑥 𝑥 ∈ 𝐴)) | |
5 | reu5 3327 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 ⊤ ↔ (∃𝑥 ∈ 𝐴 ⊤ ∧ ∃*𝑥 ∈ 𝐴 ⊤)) | |
6 | 3, 4, 5 | 3bitr4i 306 | 1 ⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃!𝑥 ∈ 𝐴 ⊤) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ⊤wtru 1544 ∃wex 1787 ∈ wcel 2112 ∃*wmo 2537 ∃!weu 2567 ∃wrex 3052 ∃!wreu 3053 ∃*wrmo 3054 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-mo 2539 df-eu 2568 df-rex 3057 df-reu 3058 df-rmo 3059 |
This theorem is referenced by: (None) |
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