Theorem rankssb 9871

Description: The subset relation is inherited by the rank function. Exercise 1 of [TakeutiZaring] p. 80. (Contributed by NM, 25-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.)

Assertion

Ref	Expression
rankssb	⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → (𝐴 ⊆ 𝐵 → (rank‘𝐴) ⊆ (rank‘𝐵)))

Proof of Theorem rankssb

Step	Hyp	Ref	Expression
1		simpr 484	. . . 4 ⊢ ((𝐵 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ 𝐵)
2		r1rankidb 9827	. . . . 5 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → 𝐵 ⊆ (𝑅1‘(rank‘𝐵)))
3	2	adantr 480	. . . 4 ⊢ ((𝐵 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ⊆ 𝐵) → 𝐵 ⊆ (𝑅1‘(rank‘𝐵)))
4	1, 3	sstrd 3990	. . 3 ⊢ ((𝐵 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ (𝑅1‘(rank‘𝐵)))
5		sswf 9831	. . . 4 ⊢ ((𝐵 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On))
6		rankdmr1 9824	. . . 4 ⊢ (rank‘𝐵) ∈ dom 𝑅1
7		rankr1bg 9826	. . . 4 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝐵) ∈ dom 𝑅1) → (𝐴 ⊆ (𝑅1‘(rank‘𝐵)) ↔ (rank‘𝐴) ⊆ (rank‘𝐵)))
8	5, 6, 7	sylancl 585	. . 3 ⊢ ((𝐵 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ⊆ 𝐵) → (𝐴 ⊆ (𝑅1‘(rank‘𝐵)) ↔ (rank‘𝐴) ⊆ (rank‘𝐵)))
9	4, 8	mpbid 231	. 2 ⊢ ((𝐵 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ⊆ 𝐵) → (rank‘𝐴) ⊆ (rank‘𝐵))
10	9	ex 412	1 ⊢ (𝐵 ∈ ∪ (𝑅1 “ On) → (𝐴 ⊆ 𝐵 → (rank‘𝐴) ⊆ (rank‘𝐵)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∈ wcel 2099   ⊆ wss 3947  ∪ cuni 4908  dom cdm 5678   “ cima 5681  Oncon0 6369  ‘cfv 6548  𝑅₁cr1 9785  rankcrnk 9786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-om 7871  df-2nd 7994  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-r1 9787  df-rank 9788

This theorem is referenced by:  rankss  9872  rankunb  9873  rankuni2b  9876  rankr1id  9885