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Mirrors > Home > MPE Home > Th. List > axprALT | Structured version Visualization version GIF version |
Description: Alternate proof of axpr 5326. (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 5319 | . . 3 ⊢ {𝑥, 𝑦} ∈ V | |
2 | 1 | isseti 3507 | . 2 ⊢ ∃𝑧 𝑧 = {𝑥, 𝑦} |
3 | dfcleq 2814 | . . 3 ⊢ (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦})) | |
4 | vex 3496 | . . . . . . 7 ⊢ 𝑤 ∈ V | |
5 | 4 | elpr 4587 | . . . . . 6 ⊢ (𝑤 ∈ {𝑥, 𝑦} ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)) |
6 | 5 | bibi2i 340 | . . . . 5 ⊢ ((𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}) ↔ (𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))) |
7 | biimpr 222 | . . . . 5 ⊢ ((𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)) → ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)) | |
8 | 6, 7 | sylbi 219 | . . . 4 ⊢ ((𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}) → ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)) |
9 | 8 | alimi 1811 | . . 3 ⊢ (∀𝑤(𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}) → ∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)) |
10 | 3, 9 | sylbi 219 | . 2 ⊢ (𝑧 = {𝑥, 𝑦} → ∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)) |
11 | 2, 10 | eximii 1836 | 1 ⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∨ wo 843 ∀wal 1534 = wceq 1536 ∃wex 1779 ∈ wcel 2113 {cpr 4566 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5187 ax-sep 5200 ax-nul 5207 ax-pow 5263 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-v 3495 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-pw 4538 df-sn 4565 df-pr 4567 |
This theorem is referenced by: (None) |
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