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Mirrors > Home > MPE Home > Th. List > Mathboxes > onmaxnelsup | Structured version Visualization version GIF version |
Description: Two ways to say the maximum element of a class of ordinals is also the supremum of that class. (Contributed by RP, 27-Jan-2025.) |
Ref | Expression |
---|---|
onmaxnelsup | ⊢ (𝐴 ⊆ On → (¬ 𝐴 ⊆ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexnal 3100 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 ¬ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ ¬ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) | |
2 | ralnex 3072 | . . . 4 ⊢ (∀𝑦 ∈ 𝐴 ¬ 𝑥 ∈ 𝑦 ↔ ¬ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) | |
3 | 2 | rexbii 3094 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ∈ 𝑦 ↔ ∃𝑥 ∈ 𝐴 ¬ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) |
4 | ssunib 41954 | . . . 4 ⊢ (𝐴 ⊆ ∪ 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) | |
5 | 4 | notbii 319 | . . 3 ⊢ (¬ 𝐴 ⊆ ∪ 𝐴 ↔ ¬ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) |
6 | 1, 3, 5 | 3bitr4ri 303 | . 2 ⊢ (¬ 𝐴 ⊆ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ∈ 𝑦) |
7 | simpl 483 | . . . . . 6 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ On) | |
8 | 7 | sselda 3981 | . . . . 5 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On) |
9 | ssel2 3976 | . . . . . 6 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On) | |
10 | 9 | adantr 481 | . . . . 5 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ On) |
11 | ontri1 6395 | . . . . 5 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝑦)) | |
12 | 8, 10, 11 | syl2anc 584 | . . . 4 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑦 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝑦)) |
13 | 12 | ralbidva 3175 | . . 3 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑥 ∈ 𝑦)) |
14 | 13 | rexbidva 3176 | . 2 ⊢ (𝐴 ⊆ On → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ∈ 𝑦)) |
15 | 6, 14 | bitr4id 289 | 1 ⊢ (𝐴 ⊆ On → (¬ 𝐴 ⊆ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 ⊆ wss 3947 ∪ cuni 4907 Oncon0 6361 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 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-br 5148 df-opab 5210 df-tr 5265 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-ord 6364 df-on 6365 |
This theorem is referenced by: (None) |
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