Theorem cardsdomelir 9992

Description: A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. This is half of the assertion cardsdomel 9993 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 9963	. . . 4 ⊢ (card‘𝐵) ∈ On
2	1	onelssi 6474	. . . 4 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐴 ⊆ (card‘𝐵))
3		ssdomg 9019	. . . 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 6918	. . . 4 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐵 ∈ dom card)
6		cardid2 9972	. . . 4 ⊢ (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
7	5, 6	syl 17	. . 3 ⊢ (𝐴 ∈ (card‘𝐵) → (card‘𝐵) ≈ 𝐵)
8		domentr 9032	. . 3 ⊢ ((𝐴 ≼ (card‘𝐵) ∧ (card‘𝐵) ≈ 𝐵) → 𝐴 ≼ 𝐵)
9	4, 7, 8	syl2anc 584	. 2 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐴 ≼ 𝐵)
10		cardne 9984	. 2 ⊢ (𝐴 ∈ (card‘𝐵) → ¬ 𝐴 ≈ 𝐵)
11		brsdom 8994	. 2 ⊢ (𝐴 ≺ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≈ 𝐵))
12	9, 10, 11	sylanbrc 583	1 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐴 ≺ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2109   ⊆ wss 3931   class class class wbr 5124  dom cdm 5659  Oncon0 6357  ‘cfv 6536   ≈ cen 8961   ≼ cdom 8962   ≺ csdm 8963  cardccrd 9954

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6360  df-on 6361  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-en 8965  df-dom 8966  df-sdom 8967  df-card 9958

This theorem is referenced by:  cardsdomel  9993  pwsdompw  10222  alephval2  10591  pwcfsdom  10602  tskcard  10800