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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 5612 | . . . . . . . . . 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 3484 | . . . . . . . . . 10 ⊢ 𝑥 ∈ V | |
| 6 | fineqv 9299 | . . . . . . . . . . 11 ⊢ (¬ ω ∈ V ↔ Fin = V) | |
| 7 | 6 | biimpi 216 | . . . . . . . . . 10 ⊢ (¬ ω ∈ V → Fin = V) |
| 8 | 5, 7 | eleqtrrid 2848 | . . . . . . . . 9 ⊢ (¬ ω ∈ V → 𝑥 ∈ Fin) |
| 9 | wofi 9325 | . . . . . . . . . 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 4006 | . . . . . . . . 9 ⊢ 𝑥 ⊆ 𝑥 | |
| 14 | wereu 5681 | . . . . . . . . . . 11 ⊢ (( < We 𝑥 ∧ (𝑥 ∈ V ∧ 𝑥 ⊆ 𝑥 ∧ 𝑥 ≠ ∅)) → ∃!𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧 < 𝑦) | |
| 15 | reurex 3384 | . . . . . . . . . . 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 1927 | . . . 4 ⊢ ((¬ ω ∈ V ∧ < Or 𝐴) → ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧 < 𝑦)) |
| 23 | df-fr 5637 | . . . 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 5639 | . . 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 1087 ∀wal 1538 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 ∃!wreu 3378 Vcvv 3480 ⊆ wss 3951 ∅c0 4333 class class class wbr 5143 Or wor 5591 Fr wfr 5634 We wwe 5636 ωcom 7887 Fincfn 8985 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 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 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 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 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-om 7888 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 |
| This theorem is referenced by: (None) |
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