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Mirrors > Home > MPE Home > Th. List > ttukeyg | Structured version Visualization version GIF version |
Description: The Teichmüller-Tukey Lemma ttukey 10508 stated with the "choice" as an antecedent (the hypothesis ∪ 𝐴 ∈ dom card says that ∪ 𝐴 is well-orderable). (Contributed by Mario Carneiro, 15-May-2015.) |
Ref | Expression |
---|---|
ttukeyg | ⊢ ((∪ 𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0 4338 | . . 3 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑧 𝑧 ∈ 𝐴) | |
2 | ttukey2g 10506 | . . . . . 6 ⊢ ((∪ 𝐴 ∈ dom card ∧ 𝑧 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥 ∈ 𝐴 (𝑧 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦)) | |
3 | simpr 484 | . . . . . . 7 ⊢ ((𝑧 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) → ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) | |
4 | 3 | reximi 3076 | . . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 (𝑧 ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) |
5 | 2, 4 | syl 17 | . . . . 5 ⊢ ((∪ 𝐴 ∈ dom card ∧ 𝑧 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) |
6 | 5 | 3exp 1116 | . . . 4 ⊢ (∪ 𝐴 ∈ dom card → (𝑧 ∈ 𝐴 → (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦))) |
7 | 6 | exlimdv 1928 | . . 3 ⊢ (∪ 𝐴 ∈ dom card → (∃𝑧 𝑧 ∈ 𝐴 → (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦))) |
8 | 1, 7 | biimtrid 241 | . 2 ⊢ (∪ 𝐴 ∈ dom card → (𝐴 ≠ ∅ → (∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦))) |
9 | 8 | 3imp 1108 | 1 ⊢ ((∪ 𝐴 ∈ dom card ∧ 𝐴 ≠ ∅ ∧ ∀𝑥(𝑥 ∈ 𝐴 ↔ (𝒫 𝑥 ∩ Fin) ⊆ 𝐴)) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ⊊ 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 ∀wal 1531 ∃wex 1773 ∈ wcel 2098 ≠ wne 2932 ∀wral 3053 ∃wrex 3062 ∩ cin 3939 ⊆ wss 3940 ⊊ wpss 3941 ∅c0 4314 𝒫 cpw 4594 ∪ cuni 4899 dom cdm 5666 Fincfn 8934 cardccrd 9925 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-om 7849 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-1o 8461 df-er 8698 df-en 8935 df-dom 8936 df-fin 8938 df-card 9929 |
This theorem is referenced by: ttukey 10508 |
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