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Mirrors > Home > MPE Home > Th. List > Mathboxes > finorwe | Structured version Visualization version GIF version |
Description: If the Axiom of Infinity is denied, every total order is a well-order. The notion of a well-order cannot be usefully expressed without the Axiom of Infinity due to the inability to quantify over proper classes. (Contributed by ML, 5-Oct-2023.) |
Ref | Expression |
---|---|
finorwe | ⊢ (¬ ω ∈ V → ( < Or 𝐴 → < We 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 484 | . . . . . . . 8 ⊢ ((¬ ω ∈ V ∧ < Or 𝐴) → ¬ ω ∈ V) | |
2 | soss 5607 | . . . . . . . . . 10 ⊢ (𝑥 ⊆ 𝐴 → ( < Or 𝐴 → < Or 𝑥)) | |
3 | 2 | com12 32 | . . . . . . . . 9 ⊢ ( < Or 𝐴 → (𝑥 ⊆ 𝐴 → < Or 𝑥)) |
4 | 3 | adantl 483 | . . . . . . . 8 ⊢ ((¬ ω ∈ V ∧ < Or 𝐴) → (𝑥 ⊆ 𝐴 → < Or 𝑥)) |
5 | vex 3479 | . . . . . . . . . 10 ⊢ 𝑥 ∈ V | |
6 | fineqv 9259 | . . . . . . . . . . 11 ⊢ (¬ ω ∈ V ↔ Fin = V) | |
7 | 6 | biimpi 215 | . . . . . . . . . 10 ⊢ (¬ ω ∈ V → Fin = V) |
8 | 5, 7 | eleqtrrid 2841 | . . . . . . . . 9 ⊢ (¬ ω ∈ V → 𝑥 ∈ Fin) |
9 | wofi 9288 | . . . . . . . . . 10 ⊢ (( < Or 𝑥 ∧ 𝑥 ∈ Fin) → < We 𝑥) | |
10 | 9 | ancoms 460 | . . . . . . . . 9 ⊢ ((𝑥 ∈ Fin ∧ < Or 𝑥) → < We 𝑥) |
11 | 8, 10 | sylan 581 | . . . . . . . 8 ⊢ ((¬ ω ∈ V ∧ < Or 𝑥) → < We 𝑥) |
12 | 1, 4, 11 | syl6an 683 | . . . . . . 7 ⊢ ((¬ ω ∈ V ∧ < Or 𝐴) → (𝑥 ⊆ 𝐴 → < We 𝑥)) |
13 | ssid 4003 | . . . . . . . . 9 ⊢ 𝑥 ⊆ 𝑥 | |
14 | wereu 5671 | . . . . . . . . . . 11 ⊢ (( < We 𝑥 ∧ (𝑥 ∈ V ∧ 𝑥 ⊆ 𝑥 ∧ 𝑥 ≠ ∅)) → ∃!𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧 < 𝑦) | |
15 | reurex 3381 | . . . . . . . . . . 11 ⊢ (∃!𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧 < 𝑦 → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧 < 𝑦) | |
16 | 14, 15 | syl 17 | . . . . . . . . . 10 ⊢ (( < We 𝑥 ∧ (𝑥 ∈ V ∧ 𝑥 ⊆ 𝑥 ∧ 𝑥 ≠ ∅)) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧 < 𝑦) |
17 | 5, 16 | mp3anr1 1459 | . . . . . . . . 9 ⊢ (( < We 𝑥 ∧ (𝑥 ⊆ 𝑥 ∧ 𝑥 ≠ ∅)) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧 < 𝑦) |
18 | 13, 17 | mpanr1 702 | . . . . . . . 8 ⊢ (( < We 𝑥 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧 < 𝑦) |
19 | 18 | ex 414 | . . . . . . 7 ⊢ ( < We 𝑥 → (𝑥 ≠ ∅ → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧 < 𝑦)) |
20 | 12, 19 | syl6 35 | . . . . . 6 ⊢ ((¬ ω ∈ V ∧ < Or 𝐴) → (𝑥 ⊆ 𝐴 → (𝑥 ≠ ∅ → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧 < 𝑦))) |
21 | 20 | impd 412 | . . . . 5 ⊢ ((¬ ω ∈ V ∧ < Or 𝐴) → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧 < 𝑦)) |
22 | 21 | alrimiv 1931 | . . . 4 ⊢ ((¬ ω ∈ V ∧ < Or 𝐴) → ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧 < 𝑦)) |
23 | df-fr 5630 | . . . 4 ⊢ ( < Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧 < 𝑦)) | |
24 | 22, 23 | sylibr 233 | . . 3 ⊢ ((¬ ω ∈ V ∧ < Or 𝐴) → < Fr 𝐴) |
25 | simpr 486 | . . 3 ⊢ ((¬ ω ∈ V ∧ < Or 𝐴) → < Or 𝐴) | |
26 | df-we 5632 | . . 3 ⊢ ( < We 𝐴 ↔ ( < Fr 𝐴 ∧ < Or 𝐴)) | |
27 | 24, 25, 26 | sylanbrc 584 | . 2 ⊢ ((¬ ω ∈ V ∧ < Or 𝐴) → < We 𝐴) |
28 | 27 | ex 414 | 1 ⊢ (¬ ω ∈ V → ( < Or 𝐴 → < We 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 ∧ w3a 1088 ∀wal 1540 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 ∃!wreu 3375 Vcvv 3475 ⊆ wss 3947 ∅c0 4321 class class class wbr 5147 Or wor 5586 Fr wfr 5627 We wwe 5629 ωcom 7850 Fincfn 8935 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-om 7851 df-1o 8461 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 |
This theorem is referenced by: (None) |
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