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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 8683 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 8653 | . . . 4 ⊢ (card‘𝐵) ∈ On | |
2 | 1 | onelssi 5753 | . . . 4 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐴 ⊆ (card‘𝐵)) |
3 | ssdomg 7887 | . . . 4 ⊢ ((card‘𝐵) ∈ On → (𝐴 ⊆ (card‘𝐵) → 𝐴 ≼ (card‘𝐵))) | |
4 | 1, 2, 3 | mpsyl 66 | . . 3 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐴 ≼ (card‘𝐵)) |
5 | elfvdm 6130 | . . . 4 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐵 ∈ dom card) | |
6 | cardid2 8662 | . . . 4 ⊢ (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝐴 ∈ (card‘𝐵) → (card‘𝐵) ≈ 𝐵) |
8 | domentr 7901 | . . 3 ⊢ ((𝐴 ≼ (card‘𝐵) ∧ (card‘𝐵) ≈ 𝐵) → 𝐴 ≼ 𝐵) | |
9 | 4, 7, 8 | syl2anc 691 | . 2 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐴 ≼ 𝐵) |
10 | cardne 8674 | . 2 ⊢ (𝐴 ∈ (card‘𝐵) → ¬ 𝐴 ≈ 𝐵) | |
11 | brsdom 7864 | . 2 ⊢ (𝐴 ≺ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≈ 𝐵)) | |
12 | 9, 10, 11 | sylanbrc 695 | 1 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐴 ≺ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 1977 ⊆ wss 3540 class class class wbr 4583 dom cdm 5038 Oncon0 5640 ‘cfv 5804 ≈ cen 7838 ≼ cdom 7839 ≺ csdm 7840 cardccrd 8644 |
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-int 4411 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-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-en 7842 df-dom 7843 df-sdom 7844 df-card 8648 |
This theorem is referenced by: cardsdomel 8683 pwsdompw 8909 alephval2 9273 pwcfsdom 9284 tskcard 9482 |
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