Theorem cardne 9394

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 6702	. 2 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐵 ∈ dom card)
2		cardon 9373	. . . . . . . . . 10 ⊢ (card‘𝐵) ∈ On
3	2	oneli 6298	. . . . . . . . 9 ⊢ (𝐴 ∈ (card‘𝐵) → 𝐴 ∈ On)
4		breq1 5069	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝐵 ↔ 𝐴 ≈ 𝐵))
5	4	onintss 6241	. . . . . . . . 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 484	. . . . . . 7 ⊢ ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → (𝐴 ≈ 𝐵 → ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐵} ⊆ 𝐴))
8		cardval3 9381	. . . . . . . . 9 ⊢ (𝐵 ∈ dom card → (card‘𝐵) = ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐵})
9	8	sseq1d 3998	. . . . . . . 8 ⊢ (𝐵 ∈ dom card → ((card‘𝐵) ⊆ 𝐴 ↔ ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐵} ⊆ 𝐴))
10	9	adantr 483	. . . . . . 7 ⊢ ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → ((card‘𝐵) ⊆ 𝐴 ↔ ∩ {𝑥 ∈ On ∣ 𝑥 ≈ 𝐵} ⊆ 𝐴))
11	7, 10	sylibrd 261	. . . . . 6 ⊢ ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → (𝐴 ≈ 𝐵 → (card‘𝐵) ⊆ 𝐴))
12		ontri1 6225	. . . . . . . 8 ⊢ (((card‘𝐵) ∈ On ∧ 𝐴 ∈ On) → ((card‘𝐵) ⊆ 𝐴 ↔ ¬ 𝐴 ∈ (card‘𝐵)))
13	2, 3, 12	sylancr 589	. . . . . . 7 ⊢ (𝐴 ∈ (card‘𝐵) → ((card‘𝐵) ⊆ 𝐴 ↔ ¬ 𝐴 ∈ (card‘𝐵)))
14	13	adantl 484	. . . . . 6 ⊢ ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → ((card‘𝐵) ⊆ 𝐴 ↔ ¬ 𝐴 ∈ (card‘𝐵)))
15	11, 14	sylibd 241	. . . . 5 ⊢ ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → (𝐴 ≈ 𝐵 → ¬ 𝐴 ∈ (card‘𝐵)))
16	15	con2d 136	. . . 4 ⊢ ((𝐵 ∈ dom card ∧ 𝐴 ∈ (card‘𝐵)) → (𝐴 ∈ (card‘𝐵) → ¬ 𝐴 ≈ 𝐵))
17	16	ex 415	. . 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 208   ∧ wa 398   ∈ wcel 2114  {crab 3142   ⊆ wss 3936  ∩ cint 4876   class class class wbr 5066  dom cdm 5555  Oncon0 6191  ‘cfv 6355   ≈ cen 8506  cardccrd 9364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-ord 6194  df-on 6195  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-fv 6363  df-en 8510  df-card 9368

This theorem is referenced by:  carden2b  9396  cardlim  9401  cardsdomelir  9402