Theorem cardne 10034

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 6957	. 2 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐵 ∈ dom card)
2		cardon 10013	. . . . . . . . . 10 ⊢ (card‘𝐵) ∈ On
3	2	oneli 6509	. . . . . . . . 9 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐴 ∈ On)
4		breq1 5169	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝐵 ↔ 𝐴 ≈ 𝐵))
5	4	onintss 6446	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (𝐴 ≈ 𝐵 → ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐵} ⊆ 𝐴))
6	3, 5	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ (card‘𝐵) → (𝐴 ≈ 𝐵 → ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐵} ⊆ 𝐴))
7	6	adantl 481	. . . . . . 7 ⊢ ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → (𝐴 ≈ 𝐵 → ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐵} ⊆ 𝐴))
8		cardval3 10021	. . . . . . . . 9 ⊢ (𝐵 ∈ dom card → (card‘𝐵) = ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐵})
9	8	sseq1d 4040	. . . . . . . 8 ⊢ (𝐵 ∈ dom card → ((card‘𝐵) ⊆ 𝐴 ↔ ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐵} ⊆ 𝐴))
10	9	adantr 480	. . . . . . 7 ⊢ ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → ((card‘𝐵) ⊆ 𝐴 ↔ ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐵} ⊆ 𝐴))
11	7, 10	sylibrd 259	. . . . . 6 ⊢ ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → (𝐴 ≈ 𝐵 → (card‘𝐵) ⊆ 𝐴))
12		ontri1 6429	. . . . . . . 8 ⊢ (((card‘𝐵) ∈ On ∧ 𝐴 ∈ On) → ((card‘𝐵) ⊆ 𝐴 ↔ ¬ 𝐴 ∈ (card‘𝐵)))
13	2, 3, 12	sylancr 586	. . . . . . 7 ⊢ (𝐴 ∈ (card‘𝐵) → ((card‘𝐵) ⊆ 𝐴 ↔ ¬ 𝐴 ∈ (card‘𝐵)))
14	13	adantl 481	. . . . . 6 ⊢ ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → ((card‘𝐵) ⊆ 𝐴 ↔ ¬ 𝐴 ∈ (card‘𝐵)))
15	11, 14	sylibd 239	. . . . 5 ⊢ ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → (𝐴 ≈ 𝐵 → ¬ 𝐴 ∈ (card‘𝐵)))
16	15	con2d 134	. . . 4 ⊢ ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → (𝐴 ∈ (card‘𝐵) → ¬ 𝐴 ≈ 𝐵))
17	16	ex 412	. . 3 ⊢ (𝐵 ∈ dom card → (𝐴 ∈ (card‘𝐵) → (𝐴 ∈ (card‘𝐵) → ¬ 𝐴 ≈ 𝐵)))
18	17	pm2.43d 53	. 2 ⊢ (𝐵 ∈ dom card → (𝐴 ∈ (card‘𝐵) → ¬ 𝐴 ≈ 𝐵))
19	1, 18	mpcom 38	1 ⊢ (𝐴 ∈ (card‘𝐵) → ¬ 𝐴 ≈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2108  {crab 3443   ⊆ wss 3976  ∩ cint 4970   class class class wbr 5166  dom cdm 5700  Oncon0 6395  ‘cfv 6573   ≈ cen 9000  cardccrd 10004

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-en 9004  df-card 10008

This theorem is referenced by:  carden2b  10036  cardlim  10041  cardsdomelir  10042