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Mirrors > Home > MPE Home > Th. List > enp1ilem | Structured version Visualization version GIF version |
Description: Lemma for uses of enp1i 8737. (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 4083 | . . 3 ⊢ ((𝐴 ∖ {𝑥}) = 𝑆 → ((𝐴 ∖ {𝑥}) ∪ {𝑥}) = (𝑆 ∪ {𝑥})) | |
2 | undif1 4382 | . . 3 ⊢ ((𝐴 ∖ {𝑥}) ∪ {𝑥}) = (𝐴 ∪ {𝑥}) | |
3 | uncom 4080 | . . . 4 ⊢ (𝑆 ∪ {𝑥}) = ({𝑥} ∪ 𝑆) | |
4 | enp1ilem.1 | . . . 4 ⊢ 𝑇 = ({𝑥} ∪ 𝑆) | |
5 | 3, 4 | eqtr4i 2824 | . . 3 ⊢ (𝑆 ∪ {𝑥}) = 𝑇 |
6 | 1, 2, 5 | 3eqtr3g 2856 | . 2 ⊢ ((𝐴 ∖ {𝑥}) = 𝑆 → (𝐴 ∪ {𝑥}) = 𝑇) |
7 | snssi 4701 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ 𝐴) | |
8 | ssequn2 4110 | . . . 4 ⊢ ({𝑥} ⊆ 𝐴 ↔ (𝐴 ∪ {𝑥}) = 𝐴) | |
9 | 7, 8 | sylib 221 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐴 ∪ {𝑥}) = 𝐴) |
10 | 9 | eqeq1d 2800 | . 2 ⊢ (𝑥 ∈ 𝐴 → ((𝐴 ∪ {𝑥}) = 𝑇 ↔ 𝐴 = 𝑇)) |
11 | 6, 10 | syl5ib 247 | 1 ⊢ (𝑥 ∈ 𝐴 → ((𝐴 ∖ {𝑥}) = 𝑆 → 𝐴 = 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ∖ cdif 3878 ∪ cun 3879 ⊆ wss 3881 {csn 4525 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-sn 4526 |
This theorem is referenced by: en2 8738 en3 8739 en4 8740 |
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