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Theorem aiota0def 43646
 Description: Example for a defined alternate 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). This corresponds to iota0def 43625. (Contributed by AV, 25-Aug-2022.)
Assertion
Ref Expression
aiota0def (℩'𝑥𝑦 𝑥𝑦) = ∅
Distinct variable group:   𝑥,𝑦

Proof of Theorem aiota0def
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 0ex 5175 . 2 ∅ ∈ V
2 al0ssb 5176 . . 3 (∀𝑦 𝑥𝑦𝑥 = ∅)
32ax-gen 1797 . 2 𝑥(∀𝑦 𝑥𝑦𝑥 = ∅)
4 eqeq2 2810 . . . . . 6 (𝑧 = ∅ → (𝑥 = 𝑧𝑥 = ∅))
54bibi2d 346 . . . . 5 (𝑧 = ∅ → ((∀𝑦 𝑥𝑦𝑥 = 𝑧) ↔ (∀𝑦 𝑥𝑦𝑥 = ∅)))
65albidv 1921 . . . 4 (𝑧 = ∅ → (∀𝑥(∀𝑦 𝑥𝑦𝑥 = 𝑧) ↔ ∀𝑥(∀𝑦 𝑥𝑦𝑥 = ∅)))
7 eqeq2 2810 . . . 4 (𝑧 = ∅ → ((℩'𝑥𝑦 𝑥𝑦) = 𝑧 ↔ (℩'𝑥𝑦 𝑥𝑦) = ∅))
86, 7imbi12d 348 . . 3 (𝑧 = ∅ → ((∀𝑥(∀𝑦 𝑥𝑦𝑥 = 𝑧) → (℩'𝑥𝑦 𝑥𝑦) = 𝑧) ↔ (∀𝑥(∀𝑦 𝑥𝑦𝑥 = ∅) → (℩'𝑥𝑦 𝑥𝑦) = ∅)))
9 aiotaval 43645 . . 3 (∀𝑥(∀𝑦 𝑥𝑦𝑥 = 𝑧) → (℩'𝑥𝑦 𝑥𝑦) = 𝑧)
108, 9vtoclg 3515 . 2 (∅ ∈ V → (∀𝑥(∀𝑦 𝑥𝑦𝑥 = ∅) → (℩'𝑥𝑦 𝑥𝑦) = ∅))
111, 3, 10mp2 9 1 (℩'𝑥𝑦 𝑥𝑦) = ∅
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536   = wceq 1538   ∈ wcel 2111  Vcvv 3441   ⊆ wss 3881  ∅c0 4243  ℩'caiota 43635 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-nul 5174 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-sn 4526  df-pr 4528  df-uni 4801  df-int 4839  df-iota 6283  df-aiota 43637 This theorem is referenced by: (None)
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