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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 34922. Note that this alternate proof uses the fact that ℩𝑥𝜑 evaluates to ∅ when there is no 𝑥 satisfying 𝜑 (iotanul 6347). 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 4301 | . 2 ⊢ ∅ ⊆ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) | |
2 | iotassuni 6348 | . . 3 ⊢ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) ⊆ ∪ {𝑥 ∣ ∀𝑦 ¬ 𝑦 ∈ 𝑥} | |
3 | eq0 4248 | . . . . . . 7 ⊢ (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦 ∈ 𝑥) | |
4 | 3 | bicomi 227 | . . . . . 6 ⊢ (∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ 𝑥 = ∅) |
5 | 4 | abbii 2804 | . . . . 5 ⊢ {𝑥 ∣ ∀𝑦 ¬ 𝑦 ∈ 𝑥} = {𝑥 ∣ 𝑥 = ∅} |
6 | 5 | unieqi 4822 | . . . 4 ⊢ ∪ {𝑥 ∣ ∀𝑦 ¬ 𝑦 ∈ 𝑥} = ∪ {𝑥 ∣ 𝑥 = ∅} |
7 | df-sn 4532 | . . . . . 6 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅} | |
8 | 7 | eqcomi 2743 | . . . . 5 ⊢ {𝑥 ∣ 𝑥 = ∅} = {∅} |
9 | 8 | unieqi 4822 | . . . 4 ⊢ ∪ {𝑥 ∣ 𝑥 = ∅} = ∪ {∅} |
10 | 0ex 5189 | . . . . 5 ⊢ ∅ ∈ V | |
11 | 10 | unisn 4831 | . . . 4 ⊢ ∪ {∅} = ∅ |
12 | 6, 9, 11 | 3eqtri 2766 | . . 3 ⊢ ∪ {𝑥 ∣ ∀𝑦 ¬ 𝑦 ∈ 𝑥} = ∅ |
13 | 2, 12 | sseqtri 3927 | . 2 ⊢ (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) ⊆ ∅ |
14 | 1, 13 | eqssi 3907 | 1 ⊢ ∅ = (℩𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∀wal 1541 = wceq 1543 {cab 2712 ∅c0 4227 {csn 4531 ∪ cuni 4809 ℩cio 6325 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-nul 5188 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-ral 3059 df-rex 3060 df-v 3403 df-sbc 3688 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-sn 4532 df-pr 4534 df-uni 4810 df-iota 6327 |
This theorem is referenced by: (None) |
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