r1ord3 - Metamath Proof Explorer

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Theorem r1ord3 9540

Description: Ordering relation for the cumulative hierarchy of sets. Part of Theorem 3.3(i) of [BellMachover] p. 478. (Contributed by NM, 22-Sep-2003.)

**Assertion**
Ref	Expression
r1ord3	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → (𝑅₁‘𝐴) ⊆ (𝑅₁‘𝐵)))

**Proof of Theorem r1ord3**
Step	Hyp	Ref	Expression
1		r1fnon 9525	. . . 4 ⊢ 𝑅₁ Fn On
2	1	fndmi 6537	. . 3 ⊢ dom 𝑅₁ = On
3	2	eleq2i 2830	. 2 ⊢ (𝐴 ∈ dom 𝑅₁ ↔ 𝐴 ∈ On)
4	2	eleq2i 2830	. 2 ⊢ (𝐵 ∈ dom 𝑅₁ ↔ 𝐵 ∈ On)
5		r1ord3g 9537	. 2 ⊢ ((𝐴 ∈ dom 𝑅₁ ∧ 𝐵 ∈ dom 𝑅₁) → (𝐴 ⊆ 𝐵 → (𝑅₁‘𝐴) ⊆ (𝑅₁‘𝐵)))
6	3, 4, 5	syl2anbr 599	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → (𝑅₁‘𝐴) ⊆ (𝑅₁‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 ⊆ wss 3887 dom cdm 5589 Oncon0 6266 ‘cfv 6433 𝑅₁cr1 9520

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-om 7713 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-r1 9522

This theorem is referenced by: bndrank 9599 rankval4 9625 aomclem2 40880

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