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Mirrors > Home > MPE Home > Th. List > ordsseleq | Structured version Visualization version GIF version |
Description: For ordinal classes, inclusion is equivalent to membership or equality. (Contributed by NM, 25-Nov-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
ordsseleq | ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sspss 4004 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵)) | |
2 | ordelpss 6230 | . . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ 𝐴 ⊊ 𝐵)) | |
3 | 2 | orbi1d 917 | . 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵) ↔ (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵))) |
4 | 1, 3 | bitr4id 293 | 1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 847 = wceq 1543 ∈ wcel 2110 ⊆ wss 3857 ⊊ wpss 3858 Ord word 6201 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-11 2158 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pr 5311 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2713 df-cleq 2726 df-clel 2812 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-br 5044 df-opab 5106 df-tr 5151 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-ord 6205 |
This theorem is referenced by: ordtri3or 6234 ordtri1 6235 ordtri2 6237 onsseleq 6243 ordsssuc 6288 ordsson 7556 ordsucelsuc 7590 limom 7649 onfununi 8067 cfslbn 9864 noextenddif 33565 finxpsuclem 35262 |
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