Theorem cardsdomelir 8799

Description: A cardinal strictly dominates its members. Equivalent to Proposition 10.37 of [TakeutiZaring] p. 93. This is half of the assertion cardsdomel 8800 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 8770	. . . 4 ⊢ (card‘𝐵) ∈ On
2	1	onelssi 5836	. . . 4 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐴 ⊆ (card‘𝐵))
3		ssdomg 8001	. . . 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 6220	. . . 4 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐵 ∈ dom card)
6		cardid2 8779	. . . 4 ⊢ (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
7	5, 6	syl 17	. . 3 ⊢ (𝐴 ∈ (card‘𝐵) → (card‘𝐵) ≈ 𝐵)
8		domentr 8015	. . 3 ⊢ ((𝐴 ≼ (card‘𝐵) ∧ (card‘𝐵) ≈ 𝐵) → 𝐴 ≼ 𝐵)
9	4, 7, 8	syl2anc 693	. 2 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐴 ≼ 𝐵)
10		cardne 8791	. 2 ⊢ (𝐴 ∈ (card‘𝐵) → ¬ 𝐴 ≈ 𝐵)
11		brsdom 7978	. 2 ⊢ (𝐴 ≺ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≈ 𝐵))
12	9, 10, 11	sylanbrc 698	1 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐴 ≺ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 1990   ⊆ wss 3574   class class class wbr 4653  dom cdm 5114  Oncon0 5723  ‘cfv 5888   ≈ cen 7952   ≼ cdom 7953   ≺ csdm 7954  cardccrd 8761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-en 7956  df-dom 7957  df-sdom 7958  df-card 8765

This theorem is referenced by:  cardsdomel  8800  pwsdompw  9026  alephval2  9394  pwcfsdom  9405  tskcard  9603