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Mirrors > Home > MPE Home > Th. List > enp1ilem | Structured version Visualization version GIF version |
Description: Lemma for uses of enp1i 9230. (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 4121 | . . 3 ⊢ ((𝐴 ∖ {𝑥}) = 𝑆 → ((𝐴 ∖ {𝑥}) ∪ {𝑥}) = (𝑆 ∪ {𝑥})) | |
2 | undif1 4440 | . . 3 ⊢ ((𝐴 ∖ {𝑥}) ∪ {𝑥}) = (𝐴 ∪ {𝑥}) | |
3 | uncom 4118 | . . . 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 4773 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ 𝐴) | |
8 | ssequn2 4148 | . . . 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 1542 ∈ wcel 2107 ∖ cdif 3912 ∪ cun 3913 ⊆ wss 3915 {csn 4591 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-sn 4592 |
This theorem is referenced by: en2 9232 en3 9233 en4 9234 |
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