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Mirrors > Home > MPE Home > Th. List > Mathboxes > onintopsscon | Structured version Visualization version GIF version |
Description: An ordinal topology is connected, expressed in constants. (Contributed by Chen-Pang He, 16-Oct-2015.) |
Ref | Expression |
---|---|
onintopsscon | ⊢ (On ∩ Top) ⊆ Con |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3758 | . . 3 ⊢ (𝑥 ∈ (On ∩ Top) ↔ (𝑥 ∈ On ∧ 𝑥 ∈ Top)) | |
2 | eloni 5650 | . . . . 5 ⊢ (𝑥 ∈ On → Ord 𝑥) | |
3 | ordtopcon 31608 | . . . . 5 ⊢ (Ord 𝑥 → (𝑥 ∈ Top ↔ 𝑥 ∈ Con)) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ (𝑥 ∈ On → (𝑥 ∈ Top ↔ 𝑥 ∈ Con)) |
5 | 4 | biimpa 500 | . . 3 ⊢ ((𝑥 ∈ On ∧ 𝑥 ∈ Top) → 𝑥 ∈ Con) |
6 | 1, 5 | sylbi 206 | . 2 ⊢ (𝑥 ∈ (On ∩ Top) → 𝑥 ∈ Con) |
7 | 6 | ssriv 3572 | 1 ⊢ (On ∩ Top) ⊆ Con |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 ∈ wcel 1977 ∩ cin 3539 ⊆ wss 3540 Ord word 5639 Oncon0 5640 Topctop 20517 Conccon 21024 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-ord 5643 df-on 5644 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-fv 5812 df-topgen 15927 df-top 20521 df-bases 20522 df-cld 20633 df-con 21025 |
This theorem is referenced by: (None) |
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