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Mirrors > Home > MPE Home > Th. List > alephsdom | Structured version Visualization version GIF version |
Description: If an ordinal is smaller than an initial ordinal, it is strictly dominated by it. (Contributed by Jeff Hankins, 24-Oct-2009.) (Proof shortened by Mario Carneiro, 20-Sep-2014.) |
Ref | Expression |
---|---|
alephsdom | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ (ℵ‘𝐵) ↔ 𝐴 ≺ (ℵ‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 485 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On) | |
2 | alephon 9488 | . . . 4 ⊢ (ℵ‘𝐵) ∈ On | |
3 | onenon 9371 | . . . 4 ⊢ ((ℵ‘𝐵) ∈ On → (ℵ‘𝐵) ∈ dom card) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ (ℵ‘𝐵) ∈ dom card |
5 | cardsdomel 9396 | . . 3 ⊢ ((𝐴 ∈ On ∧ (ℵ‘𝐵) ∈ dom card) → (𝐴 ≺ (ℵ‘𝐵) ↔ 𝐴 ∈ (card‘(ℵ‘𝐵)))) | |
6 | 1, 4, 5 | sylancl 588 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ≺ (ℵ‘𝐵) ↔ 𝐴 ∈ (card‘(ℵ‘𝐵)))) |
7 | alephcard 9489 | . . 3 ⊢ (card‘(ℵ‘𝐵)) = (ℵ‘𝐵) | |
8 | 7 | eleq2i 2903 | . 2 ⊢ (𝐴 ∈ (card‘(ℵ‘𝐵)) ↔ 𝐴 ∈ (ℵ‘𝐵)) |
9 | 6, 8 | syl6rbb 290 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ (ℵ‘𝐵) ↔ 𝐴 ≺ (ℵ‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2113 class class class wbr 5059 dom cdm 5548 Oncon0 6184 ‘cfv 6348 ≺ csdm 8501 cardccrd 9357 ℵcale 9358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-inf2 9097 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7107 df-om 7574 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-oi 8967 df-har 9015 df-card 9361 df-aleph 9362 |
This theorem is referenced by: alephdom2 9506 |
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