| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > enp1ilem | Structured version Visualization version GIF version | ||
| Description: Lemma for uses of enp1i 9313. (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 4161 | . . 3 ⊢ ((𝐴 ∖ {𝑥}) = 𝑆 → ((𝐴 ∖ {𝑥}) ∪ {𝑥}) = (𝑆 ∪ {𝑥})) | |
| 2 | undif1 4476 | . . 3 ⊢ ((𝐴 ∖ {𝑥}) ∪ {𝑥}) = (𝐴 ∪ {𝑥}) | |
| 3 | uncom 4158 | . . . 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 4808 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ 𝐴) | |
| 8 | ssequn2 4189 | . . . 4 ⊢ ({𝑥} ⊆ 𝐴 ↔ (𝐴 ∪ {𝑥}) = 𝐴) | |
| 9 | 7, 8 | sylib 218 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐴 ∪ {𝑥}) = 𝐴) |
| 10 | 9 | eqeq1d 2739 | . 2 ⊢ (𝑥 ∈ 𝐴 → ((𝐴 ∪ {𝑥}) = 𝑇 ↔ 𝐴 = 𝑇)) |
| 11 | 6, 10 | imbitrid 244 | 1 ⊢ (𝑥 ∈ 𝐴 → ((𝐴 ∖ {𝑥}) = 𝑆 → 𝐴 = 𝑇)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∖ cdif 3948 ∪ cun 3949 ⊆ wss 3951 {csn 4626 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-sn 4627 |
| This theorem is referenced by: en2 9315 en3 9316 en4 9317 |
| Copyright terms: Public domain | W3C validator |