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Mirrors > Home > MPE Home > Th. List > Mathboxes > iota0def | Structured version Visualization version GIF version |
Description: Example for a defined iota being the empty set, i.e., ∀𝑦𝑥 ⊆ 𝑦 is a wff satisfied by a unique value 𝑥, namely 𝑥 = ∅ (the empty set is the one and only set which is a subset of every set). (Contributed by AV, 24-Aug-2022.) |
Ref | Expression |
---|---|
iota0def | ⊢ (℩𝑥∀𝑦 𝑥 ⊆ 𝑦) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5213 | . 2 ⊢ ∅ ∈ V | |
2 | al0ssb 5214 | . . . 4 ⊢ (∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = ∅) | |
3 | 2 | a1i 11 | . . 3 ⊢ ((∅ ∈ V ∧ ∅ ∈ V) → (∀𝑦 𝑥 ⊆ 𝑦 ↔ 𝑥 = ∅)) |
4 | 3 | iota5 6340 | . 2 ⊢ ((∅ ∈ V ∧ ∅ ∈ V) → (℩𝑥∀𝑦 𝑥 ⊆ 𝑦) = ∅) |
5 | 1, 1, 4 | mp2an 690 | 1 ⊢ (℩𝑥∀𝑦 𝑥 ⊆ 𝑦) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∀wal 1535 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ⊆ wss 3938 ∅c0 4293 ℩cio 6314 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-nul 5212 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-sn 4570 df-pr 4572 df-uni 4841 df-iota 6316 |
This theorem is referenced by: (None) |
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