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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 36241. Note that this alternate proof uses the fact that ℩𝑥𝜑 evaluates to ∅ when there is no 𝑥 satisfying 𝜑 (iotanul 6520). 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 4395 | . 2 ⊢ ∅ ⊆ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) | |
2 | iotassuni 6514 | . . 3 ⊢ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) ⊆ ∪ {𝑥 ∣ ∀𝑦 ¬ 𝑦 ∈ 𝑥} | |
3 | eq0 4342 | . . . . . . 7 ⊢ (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ 𝑥) | |
4 | 3 | bicomi 223 | . . . . . 6 ⊢ (∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ 𝑥 = ∅) |
5 | 4 | abbii 2800 | . . . . 5 ⊢ {𝑥 ∣ ∀𝑦 ¬ 𝑦 ∈ 𝑥} = {𝑥 ∣ 𝑥 = ∅} |
6 | 5 | unieqi 4920 | . . . 4 ⊢ ∪ {𝑥 ∣ ∀𝑦 ¬ 𝑦 ∈ 𝑥} = ∪ {𝑥 ∣ 𝑥 = ∅} |
7 | df-sn 4628 | . . . . . 6 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅} | |
8 | 7 | eqcomi 2739 | . . . . 5 ⊢ {𝑥 ∣ 𝑥 = ∅} = {∅} |
9 | 8 | unieqi 4920 | . . . 4 ⊢ ∪ {𝑥 ∣ 𝑥 = ∅} = ∪ {∅} |
10 | 0ex 5306 | . . . . 5 ⊢ ∅ ∈ V | |
11 | 10 | unisn 4929 | . . . 4 ⊢ ∪ {∅} = ∅ |
12 | 6, 9, 11 | 3eqtri 2762 | . . 3 ⊢ ∪ {𝑥 ∣ ∀𝑦 ¬ 𝑦 ∈ 𝑥} = ∅ |
13 | 2, 12 | sseqtri 4017 | . 2 ⊢ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) ⊆ ∅ |
14 | 1, 13 | eqssi 3997 | 1 ⊢ ∅ = (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∀wal 1537 = wceq 1539 {cab 2707 ∅c0 4321 {csn 4627 ∪ cuni 4907 ℩cio 6492 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 ax-nul 5305 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-ne 2939 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-sn 4628 df-pr 4630 df-uni 4908 df-iota 6494 |
This theorem is referenced by: (None) |
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