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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 9573 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 9543 | . . . 4 ⊢ (card‘𝐵) ∈ On | |
2 | 1 | onelssi 6311 | . . . 4 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐴 ⊆ (card‘𝐵)) |
3 | ssdomg 8663 | . . . 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 6738 | . . . 4 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐵 ∈ dom card) | |
6 | cardid2 9552 | . . . 4 ⊢ (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝐴 ∈ (card‘𝐵) → (card‘𝐵) ≈ 𝐵) |
8 | domentr 8676 | . . 3 ⊢ ((𝐴 ≼ (card‘𝐵) ∧ (card‘𝐵) ≈ 𝐵) → 𝐴 ≼ 𝐵) | |
9 | 4, 7, 8 | syl2anc 587 | . 2 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐴 ≼ 𝐵) |
10 | cardne 9564 | . 2 ⊢ (𝐴 ∈ (card‘𝐵) → ¬ 𝐴 ≈ 𝐵) | |
11 | brsdom 8640 | . 2 ⊢ (𝐴 ≺ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≈ 𝐵)) | |
12 | 9, 10, 11 | sylanbrc 586 | 1 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐴 ≺ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2110 ⊆ wss 3857 class class class wbr 5043 dom cdm 5540 Oncon0 6202 ‘cfv 6369 ≈ cen 8612 ≼ cdom 8613 ≺ csdm 8614 cardccrd 9534 |
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-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 |
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-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-sbc 3688 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-ord 6205 df-on 6206 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-en 8616 df-dom 8617 df-sdom 8618 df-card 9538 |
This theorem is referenced by: cardsdomel 9573 pwsdompw 9801 alephval2 10169 pwcfsdom 10180 tskcard 10378 |
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