Theorem cardsdomelir 9886

Description: A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. This is half of the assertion cardsdomel 9887 and can be proven without the AC. (Contributed by Mario Carneiro, 15-Jan-2013.)

Assertion

Ref	Expression
cardsdomelir	⊢ (𝐴 ∈ (card‘𝐵) → 𝐴 ≺ 𝐵)

Proof of Theorem cardsdomelir

Step	Hyp	Ref	Expression
1		cardon 9857	. . . 4 ⊢ (card‘𝐵) ∈ On
2	1	onelssi 6428	. . . 4 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐴 ⊆ (card‘𝐵))
3		ssdomg 8936	. . . 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 6863	. . . 4 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐵 ∈ dom card)
6		cardid2 9866	. . . 4 ⊢ (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
7	5, 6	syl 17	. . 3 ⊢ (𝐴 ∈ (card‘𝐵) → (card‘𝐵) ≈ 𝐵)
8		domentr 8949	. . 3 ⊢ ((𝐴 ≼ (card‘𝐵) ∧ (card‘𝐵) ≈ 𝐵) → 𝐴 ≼ 𝐵)
9	4, 7, 8	syl2anc 585	. 2 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐴 ≼ 𝐵)
10		cardne 9878	. 2 ⊢ (𝐴 ∈ (card‘𝐵) → ¬ 𝐴 ≈ 𝐵)
11		brsdom 8910	. 2 ⊢ (𝐴 ≺ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≈ 𝐵))
12	9, 10, 11	sylanbrc 584	1 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐴 ≺ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2114   ⊆ wss 3885   class class class wbr 5074  dom cdm 5620  Oncon0 6312  ‘cfv 6487   ≈ cen 8879   ≼ cdom 8880   ≺ csdm 8881  cardccrd 9848

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-rab 3388  df-v 3429  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-int 4880  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-ord 6315  df-on 6316  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-en 8883  df-dom 8884  df-sdom 8885  df-card 9852

This theorem is referenced by:  cardsdomel  9887  pwsdompw  10114  alephval2  10484  pwcfsdom  10495  tskcard  10693