Theorem cardsdomelir 9572

Description: A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. This is half of the assertion cardsdomel 9573 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 9543	. . . 4 ⊢ (card‘𝐵) ∈ On
2	1	onelssi 6311	. . . 4 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐴 ⊆ (card‘𝐵))
3		ssdomg 8663	. . . 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 6738	. . . 4 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐵 ∈ dom card)
6		cardid2 9552	. . . 4 ⊢ (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
7	5, 6	syl 17	. . 3 ⊢ (𝐴 ∈ (card‘𝐵) → (card‘𝐵) ≈ 𝐵)
8		domentr 8676	. . 3 ⊢ ((𝐴 ≼ (card‘𝐵) ∧ (card‘𝐵) ≈ 𝐵) → 𝐴 ≼ 𝐵)
9	4, 7, 8	syl2anc 587	. 2 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐴 ≼ 𝐵)
10		cardne 9564	. 2 ⊢ (𝐴 ∈ (card‘𝐵) → ¬ 𝐴 ≈ 𝐵)
11		brsdom 8640	. 2 ⊢ (𝐴 ≺ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≈ 𝐵))
12	9, 10, 11	sylanbrc 586	1 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐴 ≺ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2110   ⊆ wss 3857   class class class wbr 5043  dom cdm 5540  Oncon0 6202  ‘cfv 6369   ≈ cen 8612   ≼ cdom 8613   ≺ csdm 8614  cardccrd 9534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2706  ax-sep 5181  ax-nul 5188  ax-pow 5247  ax-pr 5311  ax-un 7512

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2537  df-eu 2566  df-clab 2713  df-cleq 2726  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3403  df-sbc 3688  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-pss 3876  df-nul 4228  df-if 4430  df-pw 4505  df-sn 4532  df-pr 4534  df-tp 4536  df-op 4538  df-uni 4810  df-int 4850  df-br 5044  df-opab 5106  df-mpt 5125  df-tr 5151  df-id 5444  df-eprel 5449  df-po 5457  df-so 5458  df-fr 5498  df-we 5500  df-xp 5546  df-rel 5547  df-cnv 5548  df-co 5549  df-dm 5550  df-rn 5551  df-res 5552  df-ima 5553  df-ord 6205  df-on 6206  df-iota 6327  df-fun 6371  df-fn 6372  df-f 6373  df-f1 6374  df-fo 6375  df-f1o 6376  df-fv 6377  df-en 8616  df-dom 8617  df-sdom 8618  df-card 9538

This theorem is referenced by:  cardsdomel  9573  pwsdompw  9801  alephval2  10169  pwcfsdom  10180  tskcard  10378