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Mirrors > Home > MPE Home > Th. List > Mathboxes > onfisupcl | Structured version Visualization version GIF version |
Description: Sufficient condition when the supremum of a set of ordinals is the maximum element of that set. See ordunifi 9316. (Contributed by RP, 27-Jan-2025.) |
Ref | Expression |
---|---|
onfisupcl | ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ((𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∪ 𝐴 ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 765 | . . . 4 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) ∧ (𝐴 ∈ Fin ∧ 𝐴 ≠ ∅)) → 𝐴 ⊆ On) | |
2 | simprl 769 | . . . 4 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) ∧ (𝐴 ∈ Fin ∧ 𝐴 ≠ ∅)) → 𝐴 ∈ Fin) | |
3 | simprr 771 | . . . 4 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) ∧ (𝐴 ∈ Fin ∧ 𝐴 ≠ ∅)) → 𝐴 ≠ ∅) | |
4 | 1, 2, 3 | 3jca 1125 | . . 3 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) ∧ (𝐴 ∈ Fin ∧ 𝐴 ≠ ∅)) → (𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅)) |
5 | ordunifi 9316 | . . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∪ 𝐴 ∈ 𝐴) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) ∧ (𝐴 ∈ Fin ∧ 𝐴 ≠ ∅)) → ∪ 𝐴 ∈ 𝐴) |
7 | 6 | ex 411 | 1 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ∈ 𝑉) → ((𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∪ 𝐴 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 ∈ wcel 2098 ≠ wne 2930 ⊆ wss 3939 ∅c0 4318 ∪ cuni 4903 Oncon0 6364 Fincfn 8962 |
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 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-om 7869 df-en 8963 df-fin 8966 |
This theorem is referenced by: (None) |
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