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Mirrors > Home > MPE Home > Th. List > enp1ilem | Structured version Visualization version GIF version |
Description: Lemma for uses of enp1i 8080. (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 3722 | . . 3 ⊢ ((𝐴 ∖ {𝑥}) = 𝑆 → ((𝐴 ∖ {𝑥}) ∪ {𝑥}) = (𝑆 ∪ {𝑥})) | |
2 | undif1 3995 | . . 3 ⊢ ((𝐴 ∖ {𝑥}) ∪ {𝑥}) = (𝐴 ∪ {𝑥}) | |
3 | uncom 3719 | . . . 4 ⊢ (𝑆 ∪ {𝑥}) = ({𝑥} ∪ 𝑆) | |
4 | enp1ilem.1 | . . . 4 ⊢ 𝑇 = ({𝑥} ∪ 𝑆) | |
5 | 3, 4 | eqtr4i 2635 | . . 3 ⊢ (𝑆 ∪ {𝑥}) = 𝑇 |
6 | 1, 2, 5 | 3eqtr3g 2667 | . 2 ⊢ ((𝐴 ∖ {𝑥}) = 𝑆 → (𝐴 ∪ {𝑥}) = 𝑇) |
7 | snssi 4280 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ 𝐴) | |
8 | ssequn2 3748 | . . . 4 ⊢ ({𝑥} ⊆ 𝐴 ↔ (𝐴 ∪ {𝑥}) = 𝐴) | |
9 | 7, 8 | sylib 207 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐴 ∪ {𝑥}) = 𝐴) |
10 | 9 | eqeq1d 2612 | . 2 ⊢ (𝑥 ∈ 𝐴 → ((𝐴 ∪ {𝑥}) = 𝑇 ↔ 𝐴 = 𝑇)) |
11 | 6, 10 | syl5ib 233 | 1 ⊢ (𝑥 ∈ 𝐴 → ((𝐴 ∖ {𝑥}) = 𝑆 → 𝐴 = 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∖ cdif 3537 ∪ cun 3538 ⊆ wss 3540 {csn 4125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-sn 4126 |
This theorem is referenced by: en2 8081 en3 8082 en4 8083 |
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