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Mirrors > Home > MPE Home > Th. List > zfcndun | Structured version Visualization version GIF version |
Description: Axiom of Union ax-un 7228, reproved from conditionless ZFC axioms. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
zfcndun | ⊢ ∃𝑦∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axunnd 9755 | . 2 ⊢ ∃𝑦∀𝑧(∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦) | |
2 | elequ2 2121 | . . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦)) | |
3 | elequ1 2114 | . . . . . . 7 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥)) | |
4 | 2, 3 | anbi12d 624 | . . . . . 6 ⊢ (𝑤 = 𝑦 → ((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) ↔ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥))) |
5 | 4 | cbvexvw 2087 | . . . . 5 ⊢ (∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥)) |
6 | 5 | imbi1i 341 | . . . 4 ⊢ ((∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) ↔ (∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦)) |
7 | 6 | albii 1863 | . . 3 ⊢ (∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) ↔ ∀𝑧(∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦)) |
8 | 7 | exbii 1892 | . 2 ⊢ (∃𝑦∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) ↔ ∃𝑦∀𝑧(∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦)) |
9 | 1, 8 | mpbir 223 | 1 ⊢ ∃𝑦∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∀wal 1599 = wceq 1601 ∃wex 1823 ∈ wcel 2107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pr 5140 ax-un 7228 ax-reg 8788 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-br 4889 df-opab 4951 df-eprel 5268 df-fr 5316 |
This theorem is referenced by: (None) |
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