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| Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-nuliotaALT | Structured version Visualization version GIF version | ||
| Description: Alternate proof of bj-nuliota 37058. Note that this alternate proof uses the fact that ℩𝑥𝜑 evaluates to ∅ when there is no 𝑥 satisfying 𝜑 (iotanul 6539). This is an implementation detail of the encoding currently used in set.mm and should be avoided. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| bj-nuliotaALT | ⊢ ∅ = (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 0ss 4400 | . 2 ⊢ ∅ ⊆ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) | |
| 2 | iotassuni 6533 | . . 3 ⊢ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) ⊆ ∪ {𝑥 ∣ ∀𝑦 ¬ 𝑦 ∈ 𝑥} | |
| 3 | eq0 4350 | . . . . . . 7 ⊢ (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ 𝑥) | |
| 4 | 3 | bicomi 224 | . . . . . 6 ⊢ (∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ 𝑥 = ∅) | 
| 5 | 4 | abbii 2809 | . . . . 5 ⊢ {𝑥 ∣ ∀𝑦 ¬ 𝑦 ∈ 𝑥} = {𝑥 ∣ 𝑥 = ∅} | 
| 6 | 5 | unieqi 4919 | . . . 4 ⊢ ∪ {𝑥 ∣ ∀𝑦 ¬ 𝑦 ∈ 𝑥} = ∪ {𝑥 ∣ 𝑥 = ∅} | 
| 7 | df-sn 4627 | . . . . . 6 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅} | |
| 8 | 7 | eqcomi 2746 | . . . . 5 ⊢ {𝑥 ∣ 𝑥 = ∅} = {∅} | 
| 9 | 8 | unieqi 4919 | . . . 4 ⊢ ∪ {𝑥 ∣ 𝑥 = ∅} = ∪ {∅} | 
| 10 | 0ex 5307 | . . . . 5 ⊢ ∅ ∈ V | |
| 11 | 10 | unisn 4926 | . . . 4 ⊢ ∪ {∅} = ∅ | 
| 12 | 6, 9, 11 | 3eqtri 2769 | . . 3 ⊢ ∪ {𝑥 ∣ ∀𝑦 ¬ 𝑦 ∈ 𝑥} = ∅ | 
| 13 | 2, 12 | sseqtri 4032 | . 2 ⊢ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) ⊆ ∅ | 
| 14 | 1, 13 | eqssi 4000 | 1 ⊢ ∅ = (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ∀wal 1538 = wceq 1540 {cab 2714 ∅c0 4333 {csn 4626 ∪ cuni 4907 ℩cio 6512 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-sn 4627 df-pr 4629 df-uni 4908 df-iota 6514 | 
| This theorem is referenced by: (None) | 
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