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Mirrors > Home > MPE Home > Th. List > cardsdomelir | Structured version Visualization version GIF version |
Description: A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. This is half of the assertion cardsdomel 9195 and can be proven without the AC. (Contributed by Mario Carneiro, 15-Jan-2013.) |
Ref | Expression |
---|---|
cardsdomelir | ⊢ (𝐴 ∈ (card‘𝐵) → 𝐴 ≺ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cardon 9165 | . . . 4 ⊢ (card‘𝐵) ∈ On | |
2 | 1 | onelssi 6134 | . . . 4 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐴 ⊆ (card‘𝐵)) |
3 | ssdomg 8350 | . . . 4 ⊢ ((card‘𝐵) ∈ On → (𝐴 ⊆ (card‘𝐵) → 𝐴 ≼ (card‘𝐵))) | |
4 | 1, 2, 3 | mpsyl 68 | . . 3 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐴 ≼ (card‘𝐵)) |
5 | elfvdm 6528 | . . . 4 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐵 ∈ dom card) | |
6 | cardid2 9174 | . . . 4 ⊢ (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝐴 ∈ (card‘𝐵) → (card‘𝐵) ≈ 𝐵) |
8 | domentr 8363 | . . 3 ⊢ ((𝐴 ≼ (card‘𝐵) ∧ (card‘𝐵) ≈ 𝐵) → 𝐴 ≼ 𝐵) | |
9 | 4, 7, 8 | syl2anc 576 | . 2 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐴 ≼ 𝐵) |
10 | cardne 9186 | . 2 ⊢ (𝐴 ∈ (card‘𝐵) → ¬ 𝐴 ≈ 𝐵) | |
11 | brsdom 8327 | . 2 ⊢ (𝐴 ≺ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≈ 𝐵)) | |
12 | 9, 10, 11 | sylanbrc 575 | 1 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐴 ≺ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2051 ⊆ wss 3822 class class class wbr 4925 dom cdm 5403 Oncon0 6026 ‘cfv 6185 ≈ cen 8301 ≼ cdom 8302 ≺ csdm 8303 cardccrd 9156 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-ral 3086 df-rex 3087 df-rab 3090 df-v 3410 df-sbc 3675 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-ord 6029 df-on 6030 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-en 8305 df-dom 8306 df-sdom 8307 df-card 9160 |
This theorem is referenced by: cardsdomel 9195 pwsdompw 9422 alephval2 9790 pwcfsdom 9801 tskcard 9999 |
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