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Mirrors > Home > MPE Home > Th. List > Mathboxes > mosn | Structured version Visualization version GIF version |
Description: "At most one" element in a singleton. (Contributed by Zhi Wang, 19-Sep-2024.) |
Ref | Expression |
---|---|
mosn | ⊢ (𝐴 = {𝐵} → ∃*𝑥 𝑥 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rmosn 4724 | . . 3 ⊢ ∃*𝑥 ∈ {𝐵}⊤ | |
2 | rmotru 48652 | . . 3 ⊢ (∃*𝑥 𝑥 ∈ {𝐵} ↔ ∃*𝑥 ∈ {𝐵}⊤) | |
3 | 1, 2 | mpbir 231 | . 2 ⊢ ∃*𝑥 𝑥 ∈ {𝐵} |
4 | eleq2 2828 | . . 3 ⊢ (𝐴 = {𝐵} → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝐵})) | |
5 | 4 | mobidv 2547 | . 2 ⊢ (𝐴 = {𝐵} → (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃*𝑥 𝑥 ∈ {𝐵})) |
6 | 3, 5 | mpbiri 258 | 1 ⊢ (𝐴 = {𝐵} → ∃*𝑥 𝑥 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ⊤wtru 1538 ∈ wcel 2106 ∃*wmo 2536 ∃*wrmo 3377 {csn 4631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-v 3480 df-sbc 3792 df-dif 3966 df-nul 4340 df-sn 4632 |
This theorem is referenced by: mo0 48662 mosssn 48663 mo0sn 48664 |
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