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Mirrors > Home > MPE Home > Th. List > axprALT | Structured version Visualization version GIF version |
Description: Alternate proof of axpr 5295. (Contributed by NM, 14-Nov-2006.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
axprALT | ⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zfpair 5288 | . . 3 ⊢ {𝑥, 𝑦} ∈ V | |
2 | 1 | isseti 3413 | . 2 ⊢ ∃𝑧 𝑧 = {𝑥, 𝑦} |
3 | dfcleq 2731 | . . 3 ⊢ (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦})) | |
4 | vex 3402 | . . . . . . 7 ⊢ 𝑤 ∈ V | |
5 | 4 | elpr 4539 | . . . . . 6 ⊢ (𝑤 ∈ {𝑥, 𝑦} ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)) |
6 | 5 | bibi2i 341 | . . . . 5 ⊢ ((𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}) ↔ (𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))) |
7 | biimpr 223 | . . . . 5 ⊢ ((𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)) → ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)) | |
8 | 6, 7 | sylbi 220 | . . . 4 ⊢ ((𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}) → ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)) |
9 | 8 | alimi 1818 | . . 3 ⊢ (∀𝑤(𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}) → ∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)) |
10 | 3, 9 | sylbi 220 | . 2 ⊢ (𝑧 = {𝑥, 𝑦} → ∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)) |
11 | 2, 10 | eximii 1843 | 1 ⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∨ wo 846 ∀wal 1540 = wceq 1542 ∃wex 1786 ∈ wcel 2114 {cpr 4518 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-v 3400 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-pw 4490 df-sn 4517 df-pr 4519 |
This theorem is referenced by: (None) |
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