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Mirrors > Home > MPE Home > Th. List > zorn | Structured version Visualization version GIF version |
Description: Zorn's Lemma. If the union of every chain (with respect to inclusion) in a set belongs to the set, then the set contains a maximal element. This theorem is equivalent to the Axiom of Choice. Theorem 6M of [Enderton] p. 151. See zorn2 10553 for a version with general partial orderings. (Contributed by NM, 12-Aug-2004.) |
Ref | Expression |
---|---|
zornn0.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
zorn | ⊢ (∀𝑧((𝑧 ⊆ 𝐴 ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zornn0.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | numth3 10517 | . . 3 ⊢ (𝐴 ∈ V → 𝐴 ∈ dom card) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ 𝐴 ∈ dom card |
4 | zorng 10551 | . 2 ⊢ ((𝐴 ∈ dom card ∧ ∀𝑧((𝑧 ⊆ 𝐴 ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴)) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) | |
5 | 3, 4 | mpan 690 | 1 ⊢ (∀𝑧((𝑧 ⊆ 𝐴 ∧ [⊊] Or 𝑧) → ∪ 𝑧 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∀wal 1537 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 Vcvv 3481 ⊆ wss 3966 ⊊ wpss 3967 ∪ cuni 4915 Or wor 5600 dom cdm 5693 [⊊] crpss 7748 cardccrd 9982 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5288 ax-sep 5305 ax-nul 5315 ax-pow 5374 ax-pr 5441 ax-un 7761 ax-ac2 10510 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3483 df-sbc 3795 df-csb 3912 df-dif 3969 df-un 3971 df-in 3973 df-ss 3983 df-pss 3986 df-nul 4343 df-if 4535 df-pw 4610 df-sn 4635 df-pr 4637 df-op 4641 df-uni 4916 df-int 4955 df-iun 5001 df-br 5152 df-opab 5214 df-mpt 5235 df-tr 5269 df-id 5587 df-eprel 5593 df-po 5601 df-so 5602 df-fr 5645 df-se 5646 df-we 5647 df-xp 5699 df-rel 5700 df-cnv 5701 df-co 5702 df-dm 5703 df-rn 5704 df-res 5705 df-ima 5706 df-pred 6329 df-ord 6395 df-on 6396 df-suc 6398 df-iota 6522 df-fun 6571 df-fn 6572 df-f 6573 df-f1 6574 df-fo 6575 df-f1o 6576 df-fv 6577 df-isom 6578 df-riota 7395 df-ov 7441 df-rpss 7749 df-2nd 8023 df-frecs 8314 df-wrecs 8345 df-recs 8419 df-en 8994 df-card 9986 df-ac 10163 |
This theorem is referenced by: alexsubALTlem2 24081 |
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