Theorem cardne 9951

Description: No member of a cardinal number of a set is equinumerous to the set. Proposition 10.6(2) of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 9-Jan-2013.)

Assertion

Ref	Expression
cardne	⊢ (𝐴 ∈ (card‘𝐵) → ¬ 𝐴 ≈ 𝐵)

Proof of Theorem cardne

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvdm 6916	. 2 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐵 ∈ dom card)
2		cardon 9930	. . . . . . . . . 10 ⊢ (card‘𝐵) ∈ On
3	2	oneli 6477	. . . . . . . . 9 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐴 ∈ On)
4		breq1 5116	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝐵 ↔ 𝐴 ≈ 𝐵))
5	4	onintss 6414	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (𝐴 ≈ 𝐵 → ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐵} ⊆ 𝐴))
6	3, 5	syl 18	. . . . . . . 8 ⊢ (𝐴 ∈ (card‘𝐵) → (𝐴 ≈ 𝐵 → ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐵} ⊆ 𝐴))
7	6	adantl 486	. . . . . . 7 ⊢ ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → (𝐴 ≈ 𝐵 → ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐵} ⊆ 𝐴))
8		cardval3 9938	. . . . . . . . 9 ⊢ (𝐵 ∈ dom card → (card‘𝐵) = ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐵})
9	8	sseq1d 3976	. . . . . . . 8 ⊢ (𝐵 ∈ dom card → ((card‘𝐵) ⊆ 𝐴 ↔ ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐵} ⊆ 𝐴))
10	9	adantr 485	. . . . . . 7 ⊢ ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → ((card‘𝐵) ⊆ 𝐴 ↔ ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐵} ⊆ 𝐴))
11	7, 10	sylibrd 262	. . . . . 6 ⊢ ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → (𝐴 ≈ 𝐵 → (card‘𝐵) ⊆ 𝐴))
12		ontri1 6396	. . . . . . . 8 ⊢ (((card‘𝐵) ∈ On ∧ 𝐴 ∈ On) → ((card‘𝐵) ⊆ 𝐴 ↔ ¬ 𝐴 ∈ (card‘𝐵)))
13	2, 3, 12	sylancr 598	. . . . . . 7 ⊢ (𝐴 ∈ (card‘𝐵) → ((card‘𝐵) ⊆ 𝐴 ↔ ¬ 𝐴 ∈ (card‘𝐵)))
14	13	adantl 486	. . . . . 6 ⊢ ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → ((card‘𝐵) ⊆ 𝐴 ↔ ¬ 𝐴 ∈ (card‘𝐵)))
15	11, 14	sylibd 242	. . . . 5 ⊢ ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → (𝐴 ≈ 𝐵 → ¬ 𝐴 ∈ (card‘𝐵)))
16	15	con2d 135	. . . 4 ⊢ ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → (𝐴 ∈ (card‘𝐵) → ¬ 𝐴 ≈ 𝐵))
17	16	ex 417	. . 3 ⊢ (𝐵 ∈ dom card → (𝐴 ∈ (card‘𝐵) → (𝐴 ∈ (card‘𝐵) → ¬ 𝐴 ≈ 𝐵)))
18	17	pm2.43d 54	. 2 ⊢ (𝐵 ∈ dom card → (𝐴 ∈ (card‘𝐵) → ¬ 𝐴 ≈ 𝐵))
19	1, 18	mpcom 39	1 ⊢ (𝐴 ∈ (card‘𝐵) → ¬ 𝐴 ≈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∈ wcel 2149  {crab 3423   ⊆ wss 3913  ∩ cint 4916   class class class wbr 5113  dom cdm 5662  Oncon0 6361  ‘cfv 6537   ≈ cen 8940  cardccrd 9921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-ord 6364  df-on 6365  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-en 8944  df-card 9925

This theorem is referenced by:  carden2b  9953  cardlim  9958  cardsdomelir  9959