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Mirrors > Home > MPE Home > Th. List > enp1ilem | Structured version Visualization version GIF version |
Description: Lemma for uses of enp1i 9181. (Contributed by Mario Carneiro, 5-Jan-2016.) |
Ref | Expression |
---|---|
enp1ilem.1 | ⊢ 𝑇 = ({𝑥} ∪ 𝑆) |
Ref | Expression |
---|---|
enp1ilem | ⊢ (𝑥 ∈ 𝐴 → ((𝐴 ∖ {𝑥}) = 𝑆 → 𝐴 = 𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uneq1 4114 | . . 3 ⊢ ((𝐴 ∖ {𝑥}) = 𝑆 → ((𝐴 ∖ {𝑥}) ∪ {𝑥}) = (𝑆 ∪ {𝑥})) | |
2 | undif1 4433 | . . 3 ⊢ ((𝐴 ∖ {𝑥}) ∪ {𝑥}) = (𝐴 ∪ {𝑥}) | |
3 | uncom 4111 | . . . 4 ⊢ (𝑆 ∪ {𝑥}) = ({𝑥} ∪ 𝑆) | |
4 | enp1ilem.1 | . . . 4 ⊢ 𝑇 = ({𝑥} ∪ 𝑆) | |
5 | 3, 4 | eqtr4i 2768 | . . 3 ⊢ (𝑆 ∪ {𝑥}) = 𝑇 |
6 | 1, 2, 5 | 3eqtr3g 2800 | . 2 ⊢ ((𝐴 ∖ {𝑥}) = 𝑆 → (𝐴 ∪ {𝑥}) = 𝑇) |
7 | snssi 4766 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ 𝐴) | |
8 | ssequn2 4141 | . . . 4 ⊢ ({𝑥} ⊆ 𝐴 ↔ (𝐴 ∪ {𝑥}) = 𝐴) | |
9 | 7, 8 | sylib 217 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐴 ∪ {𝑥}) = 𝐴) |
10 | 9 | eqeq1d 2739 | . 2 ⊢ (𝑥 ∈ 𝐴 → ((𝐴 ∪ {𝑥}) = 𝑇 ↔ 𝐴 = 𝑇)) |
11 | 6, 10 | imbitrid 243 | 1 ⊢ (𝑥 ∈ 𝐴 → ((𝐴 ∖ {𝑥}) = 𝑆 → 𝐴 = 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ∖ cdif 3905 ∪ cun 3906 ⊆ wss 3908 {csn 4584 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2715 df-cleq 2729 df-clel 2815 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-sn 4585 |
This theorem is referenced by: en2 9183 en3 9184 en4 9185 |
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