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Mirrors > Home > MPE Home > Th. List > ordunel | Structured version Visualization version GIF version |
Description: The maximum of two ordinals belongs to a third if each of them do. (Contributed by NM, 18-Sep-2006.) (Revised by Mario Carneiro, 25-Jun-2015.) |
Ref | Expression |
---|---|
ordunel | ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐵 ∪ 𝐶) ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prssi 4746 | . . 3 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → {𝐵, 𝐶} ⊆ 𝐴) | |
2 | 1 | 3adant1 1125 | . 2 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → {𝐵, 𝐶} ⊆ 𝐴) |
3 | ordelon 6208 | . . . 4 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ On) | |
4 | 3 | 3adant3 1127 | . . 3 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → 𝐵 ∈ On) |
5 | ordelon 6208 | . . 3 ⊢ ((Ord 𝐴 ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ On) | |
6 | ordunpr 7533 | . . 3 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ∪ 𝐶) ∈ {𝐵, 𝐶}) | |
7 | 4, 5, 6 | 3imp3i2an 1340 | . 2 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐵 ∪ 𝐶) ∈ {𝐵, 𝐶}) |
8 | 2, 7 | sseldd 3966 | 1 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐵 ∪ 𝐶) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1082 ∈ wcel 2108 ∪ cun 3932 ⊆ wss 3934 {cpr 4561 Ord word 6183 Oncon0 6184 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7453 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-opab 5120 df-tr 5164 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-ord 6187 df-on 6188 |
This theorem is referenced by: oaabs2 8264 dffi3 8887 unwf 9231 rankelun 9293 infxpenlem 9431 cfsmolem 9684 r1limwun 10150 wunex2 10152 |
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