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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 482 | . . . . . . . 8 ⊢ ((¬ ω ∈ V ∧ < Or 𝐴) → ¬ ω ∈ V) | |
| 2 | soss 5547 | . . . . . . . . . 10 ⊢ (𝑥 ⊆ 𝐴 → ( < Or 𝐴 → < Or 𝑥)) | |
| 3 | 2 | com12 32 | . . . . . . . . 9 ⊢ ( < Or 𝐴 → (𝑥 ⊆ 𝐴 → < Or 𝑥)) |
| 4 | 3 | adantl 481 | . . . . . . . 8 ⊢ ((¬ ω ∈ V ∧ < Or 𝐴) → (𝑥 ⊆ 𝐴 → < Or 𝑥)) |
| 5 | vex 3441 | . . . . . . . . . 10 ⊢ 𝑥 ∈ V | |
| 6 | fineqv 9158 | . . . . . . . . . . 11 ⊢ (¬ ω ∈ V ↔ Fin = V) | |
| 7 | 6 | biimpi 216 | . . . . . . . . . 10 ⊢ (¬ ω ∈ V → Fin = V) |
| 8 | 5, 7 | eleqtrrid 2840 | . . . . . . . . 9 ⊢ (¬ ω ∈ V → 𝑥 ∈ Fin) |
| 9 | wofi 9180 | . . . . . . . . . 10 ⊢ (( < Or 𝑥 ∧ 𝑥 ∈ Fin) → < We 𝑥) | |
| 10 | 9 | ancoms 458 | . . . . . . . . 9 ⊢ ((𝑥 ∈ Fin ∧ < Or 𝑥) → < We 𝑥) |
| 11 | 8, 10 | sylan 580 | . . . . . . . 8 ⊢ ((¬ ω ∈ V ∧ < Or 𝑥) → < We 𝑥) |
| 12 | 1, 4, 11 | syl6an 684 | . . . . . . 7 ⊢ ((¬ ω ∈ V ∧ < Or 𝐴) → (𝑥 ⊆ 𝐴 → < We 𝑥)) |
| 13 | ssid 3953 | . . . . . . . . 9 ⊢ 𝑥 ⊆ 𝑥 | |
| 14 | wereu 5615 | . . . . . . . . . . 11 ⊢ (( < We 𝑥 ∧ (𝑥 ∈ V ∧ 𝑥 ⊆ 𝑥 ∧ 𝑥 ≠ ∅)) → ∃!𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧 < 𝑦) | |
| 15 | reurex 3351 | . . . . . . . . . . 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 1460 | . . . . . . . . 9 ⊢ (( < We 𝑥 ∧ (𝑥 ⊆ 𝑥 ∧ 𝑥 ≠ ∅)) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧 < 𝑦) |
| 18 | 13, 17 | mpanr1 703 | . . . . . . . 8 ⊢ (( < We 𝑥 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧 < 𝑦) |
| 19 | 18 | ex 412 | . . . . . . 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 410 | . . . . 5 ⊢ ((¬ ω ∈ V ∧ < Or 𝐴) → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧 < 𝑦)) |
| 22 | 21 | alrimiv 1928 | . . . 4 ⊢ ((¬ ω ∈ V ∧ < Or 𝐴) → ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧 < 𝑦)) |
| 23 | df-fr 5572 | . . . 4 ⊢ ( < Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧 < 𝑦)) | |
| 24 | 22, 23 | sylibr 234 | . . 3 ⊢ ((¬ ω ∈ V ∧ < Or 𝐴) → < Fr 𝐴) |
| 25 | simpr 484 | . . 3 ⊢ ((¬ ω ∈ V ∧ < Or 𝐴) → < Or 𝐴) | |
| 26 | df-we 5574 | . . 3 ⊢ ( < We 𝐴 ↔ ( < Fr 𝐴 ∧ < Or 𝐴)) | |
| 27 | 24, 25, 26 | sylanbrc 583 | . 2 ⊢ ((¬ ω ∈ V ∧ < Or 𝐴) → < We 𝐴) |
| 28 | 27 | ex 412 | 1 ⊢ (¬ ω ∈ V → ( < Or 𝐴 → < We 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 ∀wal 1539 = wceq 1541 ∈ wcel 2113 ≠ wne 2929 ∀wral 3048 ∃wrex 3057 ∃!wreu 3345 Vcvv 3437 ⊆ wss 3898 ∅c0 4282 class class class wbr 5093 Or wor 5526 Fr wfr 5569 We wwe 5571 ωcom 7802 Fincfn 8875 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-om 7803 df-1o 8391 df-er 8628 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 |
| This theorem is referenced by: (None) |
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