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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 9403 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 9373 | . . . 4 ⊢ (card‘𝐵) ∈ On | |
2 | 1 | onelssi 6299 | . . . 4 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐴 ⊆ (card‘𝐵)) |
3 | ssdomg 8555 | . . . 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 6702 | . . . 4 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐵 ∈ dom card) | |
6 | cardid2 9382 | . . . 4 ⊢ (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝐴 ∈ (card‘𝐵) → (card‘𝐵) ≈ 𝐵) |
8 | domentr 8568 | . . 3 ⊢ ((𝐴 ≼ (card‘𝐵) ∧ (card‘𝐵) ≈ 𝐵) → 𝐴 ≼ 𝐵) | |
9 | 4, 7, 8 | syl2anc 586 | . 2 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐴 ≼ 𝐵) |
10 | cardne 9394 | . 2 ⊢ (𝐴 ∈ (card‘𝐵) → ¬ 𝐴 ≈ 𝐵) | |
11 | brsdom 8532 | . 2 ⊢ (𝐴 ≺ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≈ 𝐵)) | |
12 | 9, 10, 11 | sylanbrc 585 | 1 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐴 ≺ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2114 ⊆ wss 3936 class class class wbr 5066 dom cdm 5555 Oncon0 6191 ‘cfv 6355 ≈ cen 8506 ≼ cdom 8507 ≺ csdm 8508 cardccrd 9364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-ord 6194 df-on 6195 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-en 8510 df-dom 8511 df-sdom 8512 df-card 9368 |
This theorem is referenced by: cardsdomel 9403 pwsdompw 9626 alephval2 9994 pwcfsdom 10005 tskcard 10203 |
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