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Mirrors > Home > MPE Home > Th. List > axpr | Structured version Visualization version GIF version |
Description: Unabbreviated version of
the Axiom of Pairing of ZF set theory, derived
as a theorem from the other axioms.
This theorem should not be referenced by any proof. Instead, use ax-pr 4906 below so that the uses of the Axiom of Pairing can be more easily identified. (Contributed by NM, 14-Nov-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axpr | ⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zfpair 4904 | . . 3 ⊢ {𝑥, 𝑦} ∈ V | |
2 | 1 | isseti 3209 | . 2 ⊢ ∃𝑧 𝑧 = {𝑥, 𝑦} |
3 | dfcleq 2616 | . . 3 ⊢ (𝑧 = {𝑥, 𝑦} ↔ ∀𝑤(𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦})) | |
4 | vex 3203 | . . . . . . 7 ⊢ 𝑤 ∈ V | |
5 | 4 | elpr 4198 | . . . . . 6 ⊢ (𝑤 ∈ {𝑥, 𝑦} ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)) |
6 | 5 | bibi2i 327 | . . . . 5 ⊢ ((𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}) ↔ (𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))) |
7 | biimpr 210 | . . . . 5 ⊢ ((𝑤 ∈ 𝑧 ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦)) → ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)) | |
8 | 6, 7 | sylbi 207 | . . . 4 ⊢ ((𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}) → ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)) |
9 | 8 | alimi 1739 | . . 3 ⊢ (∀𝑤(𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}) → ∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)) |
10 | 3, 9 | sylbi 207 | . 2 ⊢ (𝑧 = {𝑥, 𝑦} → ∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)) |
11 | 2, 10 | eximii 1764 | 1 ⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∨ wo 383 ∀wal 1481 = wceq 1483 ∃wex 1704 ∈ wcel 1990 {cpr 4179 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-pw 4160 df-sn 4178 df-pr 4180 |
This theorem is referenced by: (None) |
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